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Large Orbits of Supersolvable Linear Groups
Journal of Algebra 215, 235᎐247 Ž1999. Article ID jabr.1998.7730, available online at http:rrwww.idealibrary.com on
Large Orbits of Supersolvable Linear Groups Thomas R. Wolf Department of Mathematics, Ohio Uni¨ ersity, Athens, Ohio 45701 E-mail: [email protected] Communicated by George Glauberman Received December 23, 1997
The study of regular orbits of linear groups plays an important role in representation theory, particularly that of solvable groups because a chief factor of a solvable group G is an irreducible G-module. If V is a G-module, recall that ¨ in V is in a regular orbit if C G Ž ¨ . s 1, i.e., the G-orbit of ¨ is as large as possible and it has size < G <. All groups considered here are finite. Furthermore, we only consider finite vector spaces, since there is otherwise always a regular orbit and because this is where the interesting applications lie. Existence of regular orbits has had applications to Brauer’s conjectures on height zero characters and block size as well as length-type problems. Even if G is nilpotent, G need not have a regular orbit Že.g., see Examples 4.5 in wMWx.. Passman wPax shows that if G is a p-group and V completely reducible, then there exists an orbit as large as < G < 1r2 by proving C G Ž x . l C G Ž y . s 1 for some x and y in V. He also uses this to prove results about Sylow intersections in solvable groups. ‘‘Added in proof’’ at the end of Passman’s paper is a remark that the techniques produce an even larger orbit. Proofs of this have been given by Passman in correspondence and by Isaacs wIsx with a slight strengthening and different technique. Specifically, Isaacs shows that if G is nilpotent and V is a completely reducible faithful G-module, then there is ¨ in V such that
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THEOREM A. If V is a faithful completely reducible G-module for a supersol¨ able group G, then there exist x and y in V such that C G Ž x . l C G Ž y . s 1. An immediate corollary is that there is G-orbit in V with size at least < G < 1r2 . We do note that Theorem A does not remain true if supersolvable is replaced by solvable. This is because there exist irreducible G-modules V with G solvable where < G < ) < V < 2 Žsee wMW, Example 3.8x.. If G is solvable and Ž< G <, < V <. s 1, then < G < - < V < 2 Žsee wMW, Theorem 3.5Žb.; Pf, Theorem 1x. and so it is possible that Theorem A extends to this case. Robinson wRox has shown that Brauer’s block size conjecture for p-solvable G Žalso known as the k Ž GV .-conjecture. can be proven if under appropriate conditions C G Ž ¨ . has a regular module on V for some ¨ g V. Our conclusion is a little weaker, namely there is an x g V such that C G Ž x . has a regular orbit in V. The supersolvable primitive linear groups are semi-linear groups Žsee Lemma 7. and these groups have no regular orbits on their respective modules. In particular, these groups and linear groups induced from them pose an obstacle to finding large orbits Že.g., see wTux.. Much of this paper goes towards understanding what happens to linear groups induced from semi-linear groups. But we will use supersolvability even more, as we will give an example of a case where Theorem A fails for a group G that is the wreath product of two supersolvable groups. Our first proposition is quite trivial, but quite essential to our work. Even when w s 0, it has meaning and use. PROPOSITION 1. Suppose that V is a faithful G-module and that V s W1 [ ⭈⭈⭈ [ Wn for subspaces Wi that are permuted by G. Suppose that wi , x i , yi g Wi and w s w 1 q ⭈⭈⭈ qwn . If x k and y k are C G Ž w .-conjugate, then x k and y k are conjugate in C H Ž w k . where H s NG ŽWk .. In particular, x k can only be C G Ž w .-conjugate to elements of one C H Ž w k .rC G ŽWk .-conjugacy class in Wk . Proof. We may assume that x k / 0. If g g C G Ž w . and x kg s y k , then Wkg s Wk and even g g C G Ž w k . l H s C H Ž w k .. Of course, C H Ž w k . s C G Ž w k . if w k / 0. The second statement follows from the first. LEMMA 2. Suppose that V is a faithful G-module and that VC s W1 [ ⭈⭈⭈ [ Wn for C-in¨ ariant subspaces Wi that are permuted by GrC. Assume that GrC is cyclic and has a faithful orbit on W1 , . . . , Wn4 . Suppose that wi g Wi and that C G Ž wi .rC G ŽWi . has at least r ) 1 regular orbits on Wi for each i. Set w s w 1 q ⭈⭈⭈ qwn and let k s < CC G Ž w .rC <. Then C G Ž w . has at least r regular orbits on V except possibly when r s n s k s 2 and C G Ž w . has exactly one regular orbit on V. Furthermore C G Ž w . has at least 3 regular orbits on V except possibly when r s 2 and Ž k, n. g Ž1, 1., Ž2, 2., Ž2, 3., Ž3, 3.4 .
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Proof. If r ) 2, then we may choose si , t i , and u i in distinct regular orbits of C G Ž wi .rC G ŽWi .. If r s 2, we just choose si and t i . If wi and w k are C G Ž w .-conjugate, then we may assume that si and sk are also C G Ž w .conjugate as well as t i and t k and also u i and u k . By Proposition 1, it follows that si is C G Ž w .-conjugate to no t k or u k ; and that no t i is C G Ž w .-conjugate to a u k . If x i , yi g si , t i , u i 4 , then x and y are in regular orbits of C C Ž w . where x s x 1 q ⭈⭈⭈ qx n and y s y 1 q ⭈⭈⭈ qyn , because each Wi is C-invariant and V is a faithful C-module. Furthermore, w and y are not conjugate under C C Ž w . unless x i s yi for all i. So C C Ž w . has at least r n regular orbits on V. If C G Ž w . : C, we are done. Letting LrC s CC G Ž w .rC ( C G Ž w .rC C Ž w ., we may assume that L ) C and k ) 1. We may assume that W1 , . . . , Wk 4 is a regular orbit of LrC and so C s NLŽW1 .. Let z s s1 q x 2 q ⭈⭈⭈ qx n with x i in a regular orbit of C G Ž wi .rC G ŽWi ., but such that x i is not C G Ž wi .-conjugate to si for 2 F i F k. From Proposition 1, it follows that C G Ž w . l C G Ž z . : NLŽW1 . s C and thus C G Ž w . l C G Ž z . s C C Ž w . l C C Ž z . s 1, with the last equality from the last paragraph. So z is in a regular orbit of C G Ž w .. Likewise, so is t 1 q x 2 q ⭈⭈⭈ qx n provided x i is in a regular orbit of C G Ž wi .rC G ŽWi . and x i is not C G Ž wi .-conjugate to t i for 2 F i F k. Thus there are at least r ny kq1 Ž r y 1. ky 1 distinct elements lying in regular orbits of C G Ž w . which lie in distinct orbits of C C Ž w .. Because
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A solvable primitive permutation group S on ⍀ has a unique minimal normal subgroup M. Furthermore, M transitively and regularly permutes the elements of ⍀ so that < ⍀ < s < M < is a prime power. For each ␣ in ⍀, we have that S s MS␣ with M l S␣ s 1 and that the actions of S␣ on M and ⍀ are permutation isomorphic. A group is supersolvable if every chief factor has prime order. If S is supersolvable, then < M < s < ⍀ < s p for a prime p and S is isomorphic to a subgroup of the semi-direct product Z p ( Z py1 that contains Z p . In particular, S␣ l S s 1 for distinct ␣ ,  g ⍀. COROLLARY 3. Suppose that V is a faithful G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the supersol¨ able factor group GrC. Suppose that wi g Wi and set w s w 1 q w 2 q ⭈⭈⭈ qwp . If C G Ž wi .rC G ŽWi . has at least r G 3 regular orbits on Wi for each i, then C G Ž w . has a regular orbit on V. Indeed, then C G Ž w . has at least r regular orbits on V except possibly when r s 3 s p and GrC ( S3 . Proof. By Lemma 2, we may assume that CC G Ž w .rC is not cyclic and thus p ) 2. Since GrC is isomorphic to a subgroup of Z p ( Z py1 , it follows that CC G Ž w .rC is transitive on the Wi . We may choose si , t i , and u i in distinct regular orbits of C G Ž wi .rC G ŽWi . and assume by Proposition 1 that no two of si , t j , or u k are ever C G Ž w .-conjugate. Now C G Ž w . l C G Ž s . for s s s1 q t 2 q u 3 q ⭈⭈⭈ qu p must stabilize both W1 and W2 and thus lie in C Žsee comments preceding this Corollary.. Thus C G Ž w . l C G Ž s . : C C Ž w . l C C Ž s . : li C C ŽWi . s 1. We have that s s s1 q t 2 q u 3 q ⭈⭈⭈ qu p lies in a regular orbit of C G Ž w ., as desired. In fact, this argument shows there exist Ž 3r . s r Ž r y 1.Ž r y 2.r6 elements of V which lie in distinct regular orbits of C G Ž w .. For r ) 3, C G Ž w . has at least r regular orbits on V. To complete the proof, we need just show that C G Ž w . has at least 3 regular orbits on V when r s 3 and 3 - p. But then we have that s, s1 q u 2 q t 3 q ⭈⭈⭈ qt p and t 1 q u 2 q s3 q ⭈⭈⭈ qs p lie in three distinct regular orbits of C G Ž w .. Gluck’s Permutation Lemma wMW, Theorem 5.6x characterizes those solvable primitive permutation groups Ž S, ⍀ . for which S does not have a regular orbit on the power set of ⍀. In all these cases, < ⍀ < - 10. An important case in the proof is where < ⍀ < is prime, i.e., S is supersolvable. We quote this. LEMMA 4 ŽGluck.. If S is a supersol¨ able primiti¨ e permutation group on ⍀, then there is a subset ⌬ : ⍀ such that Stab S Ž ⌬ . s 1 unless < ⍀ < s 3, 5, or 7.
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DEFINITION. If V is a G-module, we say that ¨ g V lies in a semi-regular G-orbit if C G Ž ¨ . has a regular orbit on V. Furthermore, we say that ¨ is in a type k semi-regular orbit if C G Ž ¨ . has at least k regular orbits on V. LEMMA 5. Suppose that V is a faithful G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the supersol¨ able factor group GrC. Suppose that w 1 , x 1 g W1 lie in distinct NG ŽW1 .rC G ŽW1 .-semi-regular orbits of types r and s. If rs ) 2 or if rs s 2 s p, then there exist u and ¨ in distinct semi-regular G-orbits of V of types and with G rs. Proof. Choose wi , x i g Wi that are G-conjugate to w 1 and x 1 , respectively. Now each wi Žrespectively, x i . is a type r Žresp. type s . semi-regular orbit of NG ŽWi .rC G ŽWi .. Also no wi is G-conjugate to an x j by Proposition 1 applied to C G Ž0.. We may assume that GrC / 1 and thus that p is prime. Assume first that p f 3, 5, 74 so that we may apply Gluck’s lemma and choose Žwithout loss of generality. m with 1 F m - p such that StabG r C 1, . . . , m4 s 1. If u s w 1 q ⭈⭈⭈ wm q x mq1 q ⭈⭈⭈ qx p , then C G Ž u. : C and C G Ž u. s C C Ž u. has at least r m s pym G rs regular orbits on V. Also C G Ž ¨ . has at least s m r pym G rs regular orbits on V where ¨ s x 1 q ⭈⭈⭈ x m q wmy1 q ⭈⭈⭈ qwp . Furthermore u and ¨ are not conjugate in G if p is odd. If p s 2, then w s w 1 q w 2 is in a semi-regular orbit of G by Lemma 2 and w is not conjugate to u by Proposition 1. Thus, we may now assume that p s 3, 5, or 7. By the hypotheses, rs ) 2. We may assume that r G s, and so r ) 2 or r s 2 s s. Now let u* s w 1 q ⭈⭈⭈ qwpy1 q x p and ¨ * s w 1 q ⭈⭈⭈ qwpy2 q x py1 q x p , so that u* and ¨ * are not G-conjugate. Now C G Ž u*.CrC and C G Ž ¨ *.CrC are both cyclic. If r s s s 2, then Lemma 2 shows that C G Ž u*. and C G Ž ¨ *. each have Žat least. two regular orbits on V. We thus assume that r ) 2. Observe that C G Ž u*. must stabilize Wp and CC G Ž u*.rC faithfully permutes W1 , . . . , Wpy14 . Lemma 2 shows that C G Ž u*.rC G Ž u*. l C G ŽW1 q ⭈⭈⭈ qWpy1 . has at least r regular orbits on W1 q ⭈⭈⭈ qWpy1. Since C G Ž u*. : NG ŽWp ., we have that C G Ž u*.rC G Ž u*. l C G ŽWp . has at least s regular orbits on Wp . Hence C G Ž u*. has at least rs regular orbits on V s W1 q ⭈⭈⭈ qWp . Also Corollary 3 shows that C G Ž w . has a regular orbit on V where w s w 1 q w 2 q ⭈⭈⭈ qwp . Because u* and w are not conjugate, the proof is complete. COROLLARY 6. Suppose that V is a faithful G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the supersol¨ able factor group GrC. Assume that C / 1 and that NG ŽW1 .rC G ŽW1 . has a regular orbit on W1. Then there exist u and ¨ in distinct semi-regular G-orbits of V of types and with G < W1 < ) 2.
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Proof. Since C / 1, indeed < W1 < ) 2. If w g W1 is in a regular orbit NG ŽW1 .rC G ŽW1 ., then w / 0 lies in a type < W1 < semi-regular of NG ŽW1 .rC G ŽW1 ., while 0 does lie in a different semi-regular orbit of NG ŽW1 .rC G ŽW1 .. Apply Lemma 5. Suppose that V is a vector space of dimension n over GF Ž q .. We may then label the elements of V by those of GF Ž q n . in a one-to-one fashion. We let ⌫ Ž V . or ⌫ Ž q n . denote the group of semi-linear transformations of V, i.e., ⌫ Ž V . s x ª ax <0 / a g GF Ž q n ., g GalŽ GF Ž q n .rGF Ž q ..4 . In particular, ⌫ Ž V . has a cyclic normal subgroup ⌫0 Ž V . of order q n y 1 that acts fixed-point-freely on V Žby multiplications. and the factor group ⌫ Ž V .r⌫0 Ž V . is cyclic of order n Žisomorphic to the Galois group.. In particular, ⌫ Ž V . is metacyclic. For a faithful G-module V, we will say G F ⌫ Ž V . if the elements of V may be labeled in a way that yields G a subgroup of ⌫ Ž V .. The semi-linear groups ⌫ Ž V . are supersolvable and play an important role in representation theory. We gather some known facts about supersolvable groups, the last of which characterizes primitive supersolvable linear groups as semi-linear groups. LEMMA 7.
Let G be supersol¨ able. Then
Ža. GrFŽ G . is abelian; Žb. G has a normal Sylow-p-subgroup if p is the largest prime di¨ isor of < G <; and Žc. If V is a faithful primiti¨ e and finite G-module, then G F ⌫ Ž V .. Proof. Parts Ža. and Žb. are immediate from the definition of supersolvability and are well known. Let A be a maximal abelian normal subgroup of G. Again, it is well known and direct from the definition of supersolvable that A s C G Ž A.. Since V is a primitive G-module and A is normal in G, VA is homogeneous by Clifford’s Theorem. Because VA is homogeneous and A s C G Ž A., it follows that G F ⌫ Ž V . Že.g., see wMW, Lemma 2.2x.. Recall, for positive integers q ) 1 and n, a prime divisor r of q n y 1 is called a Zsigmondy prime di¨ isor of q n y 1 if r does not divide q m y 1 whenever 0 - m - n. In this case, n is the order of q Žmod r . and thus n divides Ž r . s r y 1 for Euler’s -function. In particular, gcdŽ r, n. s 1. The Zsigmondy Prime Theorem asserts that q n y 1 always has a Zsigmondy prime divisor except when q n s 2 6 or when n s 2 and q is a Mersenne prime. LEMMA 8. Suppose that V is a faithful FG-module with G supersol¨ able and that G acts transiti¨ ely on V y 04 . Set < V < s q n where q s < F <. Ži .
Then G : ⌫ Ž V .; and
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Žii. If r is a Zsigmondy prime di¨ isor of q n y 1 and R g Syl r Ž G ., then R : ⌫0 Ž V . l G s FŽ G . s C G Ž R .; Žiii. If q n s 2 6 , then FŽ G . s G l ⌫0 Ž V . is cyclic of order 21 or 63. Proof. The transitivity hypothesis implies that V has no proper G-modules and is thus irreducible. Suppose that V s V1 [ ⭈⭈⭈ [ Vn for modules Vi that are permuted by G. If ¨ i g Vi is non-zero for each i, then ¨ 1 cannot be G-conjugate to ¨ 1 q ¨ 2 . Thus the transitivity hypothesis implies that n s 1 and that V is a primitive G-module. By Lemma 7Žc., G : ⌫ Ž V .. If r is a Zsigmondy prime divisor of q n y 1 and R g Syl r Ž G ., then 1 / R by transitivity and also R : ⌫0 Ž V . because gcdŽ r, n. s 1 Žsee paragraph preceding this lemma.. Statements of Žii. and Žiii. are immediate from Lemmas 6.4 and 6.5Žb., respectively, of wMWx. PROPOSITION 9. If G s ⌫ Ž V . and 0 / x g V, then C G Ž x . has at least three non-zero regular orbits on V unless < V < s 2 n with n F 3 or < V < s 3. Proof. Set < V < s q n for a prime power q. If n s 1, then C G Ž x . s 1 and the result is trivial. Now G acts transitively on V 噛 and so we may replace x by any non-zero element of V. If is a field automorphism of order n, then ² : is the centralizer of an element of V and so we may assume that C G Ž x . s ² :. Now C s CŽ x . is cyclic of order n. If n is prime, then C V Ž C . has order q and every element in V y C V Ž C . is in a regular orbit of C. Since Ž q n y q .rn G 3 unless q s 2 and n F 3, the result follows when n is prime. If n is a power of two, then C has a unique involution t. Every element of V y C V Ž t . is in a regular orbit of C. Since
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Suppose that V is faithful, irreducible G-module and that VC s W1 [ ⭈⭈⭈ [ Wn for C-invariant submodules Wi that are permuted faithfully by the factor group GrC with n ) 1. Then C / 1, since otherwise C V Ž G . / 1. Also W1 is an irreducible H-module where H s NG ŽW1 .rC G ŽW1 ., because G is isomorphic as a linear group to an irreducible subgroup of the wreath product H wrŽ GrC . Žsee wMW, Lemma 2.8x.. Because C / 1, also H / 1 and < W1 < ) 2. THEOREM 10. Suppose that V is a faithful, irreducible G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the factor group GrC / 1. If G is supersol¨ able and NG ŽW1 .rC G ŽW1 . is isomorphic to a subgroup of ⌫ ŽW1 ., then G has two distinct semi-regular orbits on V of types and with ) 2 except possibly when: Ži. p s 2, < W1 < s 2 2 , and s 2; Žii. p s 3, < W1 < s 2 2 , GrC ( S3 , and there exist x and y in V in distinct semi-regular orbits of G with y in a type 2 semi-regular orbit and such that C G Ž x . has three distinct orbits in V that lie in regular orbits of C FŽG.Ž x .; or Žiii. p s 2, < W1 < s q 2 for a Mersenne prime q, and G has a type 3 semi-regular orbit. Proof. Because the primitive permutation group GrC / 1 is supersolvable, p is prime and GrC is isomorphic to a subgroup of Z p ( Z py1. Set Hi s NG ŽWi . and set Ci s C G ŽWi . and < Wi < s q n for a prime power q s < F < where F is the underlying field. Because V is an irreducible G-module, also W1 is an irreducible H1rC1-module and < W1 < / 2 Žsee comments before the theorem.. If H1rC1 has a regular orbit on W1 , then G satisfies the principle conclusion of the theorem by Corollary 6. Thus we may assume that H1rC1 is not isomorphic to a subgroup of ⌫0 ŽW1 . and hence that n ) 1. If H1rC1 has at least two type 2 semi-regular orbits on W1 , then Lemma 5 shows that G has two distinct semi-regular orbits on V of types and with ) 2. But, by Lemma 7, every non-zero element of W1 is in a type 2 semi-regular orbit of H1rC1 except possibly when < W1 < s 4 and H1rC1 ( ⌫ Ž2 2 . ( S3 . Hence, in all cases, we may assume that H1rC1 acts transitively on the non-zero vectors of W1. We denote by Di that subgroup of Hi defined by DirCi s ⌫0 ŽWi . l HirCi . Then DirCi acts on Wi by field multiplications and Ci s C D i Ž x . for all non-zero x in Wi . Now, if < Wi < s q n for a prime power q, then DirCi and HirDi are cyclic with < DirCi < dividing q n y 1 and < HirDi < dividing n. Set D s Fis1 to p Di . Because Fis1 to p Ci s 1 and Fis1 to p Hi s C, it follows that D and CrD are abelian with exponents dividing q n y 1 and n Žrespectively.. Furthermore, if x i is a non-zero element of Wi , then C D Ž x 1 q ⭈⭈⭈ qx p . : lC D i Ž x i . : lCi s 1. Unless q n s q 2 for a
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Mersenne prime q, then DirCi s FŽ HirDi . by Lemma 8 and the Zsigmondy prime theorem. Standard arguments Žsimilar to those of wMW, Proposition 9.5Ža.x. then show that D s FŽ C . unless q n s q 2 for a Mersenne prime q. The primitive permutation group GrC has a unique minimal normal subgroup MrC of order p. We claim that CrD is central in MrD. If FŽ G . s FŽ C . s D, then GrD is abelian by Lemma 7. If FŽ G . ) FŽ C . s D, then MrD s CrD = FŽ G .rD. If p ) n, then p is the largest prime divisor of < MrD <, whence MrD has a normal Sylow-p-subgroup PrD by Proposition 9 and so MrD s CrD = PrD. The claim holds unless FŽ C . ) D and p F n. By the last paragraph, this can only happen when q is a Mersenne prime and n s 2 s p. But, in this exceptional case, G has a type 3 semi-regular orbit by Proposition 9 and Lemma 2 and thus exception Žiii. of this theorem holds. Thus we may assume that CrD is central in MrD. Now MrC transitively permutes the subgroups C l DirD, while CrD is central in MrD. Thus C l D 1 s ⭈⭈⭈ s C l Dp . But lDi s D and so D s C l D 1 s ⭈⭈⭈ s C l Dp . Now CrD is isomorphic to a subgroup of H1rD 1 and is hence cyclic. If x i g Wi is non-zero for all i, note that C D Ž x i . : Ci and thus Fi C D Ž x i . s 1. If u s u1 q ⭈⭈⭈ qu p and ¨ s ¨ 1 q ⭈⭈⭈ q¨ p with u i , ¨ i g Wi , then we claim that C C Ž u. l C C Ž ¨ . s 1 provided that u i or ¨ i is non-zero for each i and that ¨ j , for some j, is in a regular orbit of C G Ž u j .rC j . The claim is valid because C D Ž u. l C D Ž ¨ . s 1 and C C Ž u. l C C Ž ¨ . : C C Ž u j . l C C Ž ¨ j . : C j l C : Dj l C s D. If x i is a non-zero element of Wi , we let r be the number of regular orbits and s be the number of non-zero orbits of C G Ž x i .rCi on Wi . These values are independent of choices because H1rC1 acts transitively on the non-zero vectors of W1 and GrC transitively permutes the Wi . Since H1rC1 does not have a regular orbit on W1 , s ) r G 1 with the last inequality by Proposition 9. Fix non-zero x i g Wi and set y s x 1 q ⭈⭈⭈ qx py1. If z s z1 q ⭈⭈⭈ qz p with z i g Wi with z p / 0 and z py1 in a regular orbit of C G Ž x py1 .rC py1 , then z is in a regular orbit of C C Ž y . by the next to last paragraph. If, in addition, z py 1 is not conjugate to z i for all i - p y 1, then C G Ž y . l C G Ž z . must normalize Wpy 1 and Wp and thus lies in C. With this additional condition, we have that z is in a regular orbit of C G Ž y .. Furthermore a s a1 q ⭈⭈⭈ qa p with a i g Wi can only be C C Ž y .-conjugate to z if a i is C G Ž x i .rCi-conjugate to z i for i - p. Thus we have rs py 2 elements of V that lie in regular orbits of C G Ž y . and in distinct orbits of C C Ž y .. Since
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one of z i for i - 3 is in regular orbit of C G Ž x i .rCi . In this case, we have 2 ⭈ 1 ⭈ 2r2 regular orbits of C G Ž y .. Summarizing, C G Ž y . has 2 regular orbits on V except possibly when r s 1 and p s 2. When r s 1, then Proposition 9 shows that < W1 < s 2 2 and H1rC1 ( ⌫ Ž2 2 . ( S3 . Next we let x s x 1 q ⭈⭈⭈ qx p . If z s z1 q ⭈⭈⭈ qz p with z i g Wi with some z i in a regular orbit of C G Ž x i .rCi , then z is in a regular orbit of C C Ž x . by above because every x i is non-zero. Let u i g Wi be in a regular orbit of C G Ž x i .rCi , so that u i / 0 because HirCi has no regular orbit on Wi . Then u s 0 q ⭈⭈⭈ q0 q x py1 q u p and ¨ s u1 q ⭈⭈⭈ qu py2 q x py1 q 0 lie in regular orbits of C G Ž x .. If p ) 3, then u and ¨ cannot even be G-conjugate and C G Ž x . has at least two regular orbits on V. Also C G Ž x . has at least r regular orbits on V, because u and 0 q ⭈⭈⭈ q0 q x py 1 q t p can only be C G Ž x .-conjugate if t p and u p are C G Ž x p .rC p-conjugate. If GrC is cyclic, then u and 0 q ⭈⭈⭈ q0 q u p are in distinct regular orbits of C G Ž x . on V. Hence C G Ž x . has two regular orbits on V provided p ) 3, GrC is cyclic or r ) 1; i.e., x is a type 2 semi-regular orbit unless p s 3, GrC ( S3 , < W1 < s 2 2 , and H1rC1 ( ⌫ Ž2 2 . ( S3 . But do observe in this case that we do have one regular orbit of C G Ž x . on V. Observe, in this exceptional case, that the three elements 0 q x 2 q u 3 , 0 q u 2 q u 3 , and 0 q 0 q u 3 lie in distinct orbits of C G Ž x . and are in regular orbits of C FŽ G.Ž x ., because FŽ G .rD has order three and transitively permutes the Wi and C D Ž x . s 1. Now x and y cannot be G-conjugate because all the x i are non-zero. Both are in semi-regular orbits of G and at least one lies in a type 2 semi-regular orbit. Indeed both lie in type 2 semi-regular orbits of G or exception Ži. or Žii. applies. This completes the proof. We used the supersolvability of G in critical ways in Theorem 10 Žsee Example 13.. This will be used again in the next two results. LEMMA 11. Suppose that V is a faithful G-module and V s V1 [ ⭈⭈⭈ [ Vp for subspaces that are permuted primiti¨ ely by G. Suppose that Hi s NG Ž Vi . and that V1 can be written as a direct sum of q ) 1 subspaces that are permuted primiti¨ ely by H1. If p ) q and G is supersol¨ able, then V can be written as a direct sum of q ) 1 subspaces that are permuted primiti¨ ely by G. Proof. Let C be the kernel of the permutation action of G on V1 , . . . , Vp 4 . We may write V1 as a direct sum W11 [ ⭈⭈⭈ [ W1 q of subspaces primitively and faithfully permuted by H1rD 1 for a normal subgroup D 1 of H1. For g, h in the same right coset of H1 in G, observe that W11 , . . . , W1 q 4 g s W11 , . . . , W1 q 4 h. Hence, we may write Vi as a direct sum Wi1 [ ⭈⭈⭈ [ Wi q of subspaces primitively and faithfully permuted by HirDi for a normal subgroup Di of Hi that is G-conjugate to D 1. Furthermore, if we set D s lDi : lHi s C, then the group GrD permutes the set Wi j 4 of pq subspaces transitively and faithfully.
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Since GrC is supersolvable, GrC has a unique minimal normal subgroup MrC of prime order p that transitively permutes the Vi and a cyclic factor group GrM whose order divides p y 1. Now HirDi has a unique minimal normal subgroup of prime order q and a cyclic factor group whose order divides q y 1. Since lHi s C and lDi s D, it follows from the hypothesis p ) q that p must be larger than all prime divisors of < CrD 5 GrM <. By the supersolvability of G, GrD has a normal Sylow-psubgroup PrC of order p. Now PrC acts on Wi j 4 with q faithful orbits and we may write V s X 1 [ ⭈⭈⭈ [ X q for P-invariant submodules X k that are permuted transitively by GrP. If LrP is the kernel of the action of GrP on X 1 , . . . , X q 4 , then GrL even acts primitively on X 1 , . . . , X q 4 because q is prime. THEOREM 12. Suppose that V is a faithful, irreducible G-module for a supersol¨ able group G. Assume that V is an imprimiti¨ e G-module and choose p ) 1 as small as possible so that V may be written V s W1 [ ⭈⭈⭈ [ Wp for subspaces Wi that are permuted primiti¨ ely by G. If W1 is imprimiti¨ e as an NG ŽW1 .rC G ŽW1 .-module, then Ži. p s 2 and G has a type 3 semi-regular orbit; or Žii. G has two distinct semi-regular orbits of types and on V such that G min 3, p4 . Proof. Now W1 is imprimitive as an H-module where H s NG ŽW1 .rC G ŽW1 .. Choose q ) 1 as small as possible such that W1 may be written W1 s U1 [ ⭈⭈⭈ [ Uq as a direct sum of q subspaces that are permuted primitively by HrD for a normal subgroup D of H. By Lemma 11 and choice of p, it follows that q G p. First suppose that U1 is imprimitive as a NH ŽU1 .rC H ŽU1 .-module. Arguing by induction on dimŽ V ., we may conclude that: Ža. H has two distinct semi-regular orbits of types r and s on W such that rs G min 3, q4 ; or Žb. q s 2 and H has a type 3 semi-regular orbit on W. On the other hand, if U1 is a primitive NH ŽU1 .rC H ŽU1 .-module, then Lemma 7 shows NH ŽU1 .rC H ŽU1 . is isomorphic to a subgroup of ⌫ ŽU1 .. Then we apply Theorem 10 to conclude that H and W satisfy Ža. or Žb. above or that: Žc. q s 3, < U1 < s 2 2 , H is a 2, 34 -group, and there exist x and y in W1 in distinct semi-regular orbits of H with y in a type 2 semi-regular orbit and such that C H Ž x . has three distinct orbits in V that lie in regular orbits of regular C FŽ H .Ž x .. Whether U1 is primitive or imprimitive, H and W satisfy Ža., Žb., or Žc..
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If q s 2, then also p s 2 by Lemma 11 and choice of p. If H and W satisfy Žb., then p s 2 and G has a type 3 semi-regular orbit by Lemma 2. Conclusion Ži. of the theorem is met here. If H satisfies Ža., then Lemma 5 shows that G has two distinct semi-regular orbits of types and on V such that G min 3, q4 . Conclusion Žii. follows here as q G p. Similarly, Lemma 5 shows that Conclusion Žii. holds if H satisfies Žc. and p s 2. To complete the proof, we may assume that Žc. above holds and p s 3. Recalling V s W1 [ W2 [ W3 , we let Hi s NG ŽWi ., let Ci s C G ŽWi ., and C s lHi so that H s H1rC1 and lCi s 1. Let FirCi s FŽ HirCi . and observe that lFi s FŽ C .. Now GrC primitively and faithfully permutes W1 , W2 , W3 4 and is isomorphic to a transitive subgroup of S3 . Because G is a supersolvable 2, 34 -group, G has a normal Sylow-3-subgroup. But charŽ V . s 2 and V is irreducible, so that O 2 Ž G . s 1 and FŽ G . is the Sylow-3-subgroup of G. Now FŽ G .rFŽ C . is isomorphic to the Sylow-3-subgroup MrC of GrC and transitively permutes W1 , W2 , W3 4 . Also CrFŽ C . is the 2-group and so MrFŽ C . s CrFŽ C . = FŽ G .rFŽ C .. Since FŽ G .rFŽ C . transitively permutes the groups Fi l CrFŽ C . and centralizes CrFŽ C ., it follows that F1 l C s F2 l C s F3 l C s FŽ C .. By Žc. above, we may choose x i , yi g Wi in distinct semi-regular orbits in Wi of HirCi of types 1 and 2 Žrespectively. and such that C G Ž x i .rCi : HirCi has three distinct orbits on Wi that lie in regular orbits of C G Ž x i . l FirCi . We may assume that no x i and y k are G-conjugate. Suppose that si , t i g Wi lie in distinct regular orbits of C G Ž yi .rCi for each i. Also choose a i , bi , c i g Wi in distinct orbits of C G Ž x i . : Hi , each of which is in a regular orbit C G Ž x i . l FirCi . For i / j, we may assume that a i and bj are not conjugate via C G Ž x i q x j ., etc. Then C G Ž y 1 q x 2 q x 3 . l C G Ž s1 q b 2 q c 3 . : lHi s C. Now C C Ž y 1 q x 2 q x 3 . l C C Ž s1 q b 2 q c 3 . : C C Ž y 1 . l C C Ž s1 . : C1 l C : F1 l C s FŽ C .. Because FŽ C . : Fi , it follows that C FŽ C .Ž y 1 q x 2 q x 3 . l C FŽC .Ž s1 q b 2 q c 3 . : lCi s 1. Hence C G Ž y 1 q x 2 q x 3 . l C G Ž s1 q b 2 q c 3 . s 1 and so s1 q b 2 q c 3 is in a regular orbit of C G Ž y 1 q x 2 q x 3 .. Likewise, t1 q b 2 q c3 and s1 q a2 q c3 are in regular orbits of C G Ž y 1 q x 2 q x 3 .. Thus y 1 q x 2 q x 3 lies in a type 3 semi-regular orbit of G Žsee Proposition 1.. Now similar arguments show that s1 q t 2 q a3 , s1 q t 2 q b 3 , and s1 q t 2 q c 3 lie in regular orbits of C G Ž y 1 q y 2 q x 3 . and these lie in distinct orbits of C G Ž y 1 q y 2 q x 3 . by Proposition 1. Both y 1 q y 2 q x 3 and y 1 q x 2 q x 3 lie in a type 3 semi-regular orbits of G. Because y 1 q x 2 q x 3 and y 1 q y 2 q x 3 are not G-conjugate, conclusion Žii. of this theorem is met. Proof of Theorem A. Arguing by induction on dimŽ V ., we may assume that V is irreducible. If V is a primitive G-module, then G : ⌫ Ž V . by Lemma 7, whence Proposition 9 gives the existence of a semi-regular orbit.
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Otherwise we may write V s W1 [ ⭈⭈⭈ [ Wp for subspaces Wi that are permuted primitively by G and do so with p as small as possible. If W1 is imprimitive as a NG ŽW1 .rC G ŽW1 .-module, then Theorem 12 gives the existence of a semi-regular orbit. If W1 is primitive as a NG ŽW1 .rC G ŽW1 .module, then Theorem 10 gives the existence of a semi-regular orbit. We next give an example where Theorem A fails for a group G that is the wreath product H wr S of supersolvable groups acting on V s W G , but where the conclusion of Theorem A is valid for the action of H on W. While the action here is not coprime, Isaacs’ question of an orbit as large as < G < 1r2 also fails here. EXAMPLE 13. Let H s S3 s S, so that H acts irreducibly on a vector space W of order 2 2 and has exactly one semi-regular orbit. The wreath product G s H wr S of order 2 4 3 4 acts irreducibly on a vector space V of order 2 6 . The G-orbits in V have size 1, 9, 27, and 27 and so no G-orbit has size as large as < G < 1r2 . Furthermore, for each ¨ g V,
REFERENCES wIsx wMWx wPfx wPax wRox wTux
I. M. Isaacs, Large orbits in nilpotent actions, Proc. Amer. Math. Soc., in press. O. Manz and T. Wolf, ‘‘Representations of Solvable Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1993. P. P. Palfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77 Ž1982., 127᎐137. D. Passman, Groups with normal Hall-p⬘-subgroups, Trans. Amer. Math. Soc. 123 Ž1966., 99᎐111. G. Robinson, Further reductions for the k Ž GV .-problem, J. Algebra 195 Ž1997., 141᎐150. A. Turull, Supersolvable automorphism groups of solvable groups, Math. Z. 183 Ž1983., 47᎐73.
Large Orbits of Supersolvable Linear Groups Thomas R. Wolf Department of Mathematics, Ohio Uni¨ ersity, Athens, Ohio 45701 E-mail: [email protected] Communicated by George Glauberman Received December 23, 1997
The study of regular orbits of linear groups plays an important role in representation theory, particularly that of solvable groups because a chief factor of a solvable group G is an irreducible G-module. If V is a G-module, recall that ¨ in V is in a regular orbit if C G Ž ¨ . s 1, i.e., the G-orbit of ¨ is as large as possible and it has size < G <. All groups considered here are finite. Furthermore, we only consider finite vector spaces, since there is otherwise always a regular orbit and because this is where the interesting applications lie. Existence of regular orbits has had applications to Brauer’s conjectures on height zero characters and block size as well as length-type problems. Even if G is nilpotent, G need not have a regular orbit Že.g., see Examples 4.5 in wMWx.. Passman wPax shows that if G is a p-group and V completely reducible, then there exists an orbit as large as < G < 1r2 by proving C G Ž x . l C G Ž y . s 1 for some x and y in V. He also uses this to prove results about Sylow intersections in solvable groups. ‘‘Added in proof’’ at the end of Passman’s paper is a remark that the techniques produce an even larger orbit. Proofs of this have been given by Passman in correspondence and by Isaacs wIsx with a slight strengthening and different technique. Specifically, Isaacs shows that if G is nilpotent and V is a completely reducible faithful G-module, then there is ¨ in V such that
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THEOREM A. If V is a faithful completely reducible G-module for a supersol¨ able group G, then there exist x and y in V such that C G Ž x . l C G Ž y . s 1. An immediate corollary is that there is G-orbit in V with size at least < G < 1r2 . We do note that Theorem A does not remain true if supersolvable is replaced by solvable. This is because there exist irreducible G-modules V with G solvable where < G < ) < V < 2 Žsee wMW, Example 3.8x.. If G is solvable and Ž< G <, < V <. s 1, then < G < - < V < 2 Žsee wMW, Theorem 3.5Žb.; Pf, Theorem 1x. and so it is possible that Theorem A extends to this case. Robinson wRox has shown that Brauer’s block size conjecture for p-solvable G Žalso known as the k Ž GV .-conjecture. can be proven if under appropriate conditions C G Ž ¨ . has a regular module on V for some ¨ g V. Our conclusion is a little weaker, namely there is an x g V such that C G Ž x . has a regular orbit in V. The supersolvable primitive linear groups are semi-linear groups Žsee Lemma 7. and these groups have no regular orbits on their respective modules. In particular, these groups and linear groups induced from them pose an obstacle to finding large orbits Že.g., see wTux.. Much of this paper goes towards understanding what happens to linear groups induced from semi-linear groups. But we will use supersolvability even more, as we will give an example of a case where Theorem A fails for a group G that is the wreath product of two supersolvable groups. Our first proposition is quite trivial, but quite essential to our work. Even when w s 0, it has meaning and use. PROPOSITION 1. Suppose that V is a faithful G-module and that V s W1 [ ⭈⭈⭈ [ Wn for subspaces Wi that are permuted by G. Suppose that wi , x i , yi g Wi and w s w 1 q ⭈⭈⭈ qwn . If x k and y k are C G Ž w .-conjugate, then x k and y k are conjugate in C H Ž w k . where H s NG ŽWk .. In particular, x k can only be C G Ž w .-conjugate to elements of one C H Ž w k .rC G ŽWk .-conjugacy class in Wk . Proof. We may assume that x k / 0. If g g C G Ž w . and x kg s y k , then Wkg s Wk and even g g C G Ž w k . l H s C H Ž w k .. Of course, C H Ž w k . s C G Ž w k . if w k / 0. The second statement follows from the first. LEMMA 2. Suppose that V is a faithful G-module and that VC s W1 [ ⭈⭈⭈ [ Wn for C-in¨ ariant subspaces Wi that are permuted by GrC. Assume that GrC is cyclic and has a faithful orbit on W1 , . . . , Wn4 . Suppose that wi g Wi and that C G Ž wi .rC G ŽWi . has at least r ) 1 regular orbits on Wi for each i. Set w s w 1 q ⭈⭈⭈ qwn and let k s < CC G Ž w .rC <. Then C G Ž w . has at least r regular orbits on V except possibly when r s n s k s 2 and C G Ž w . has exactly one regular orbit on V. Furthermore C G Ž w . has at least 3 regular orbits on V except possibly when r s 2 and Ž k, n. g Ž1, 1., Ž2, 2., Ž2, 3., Ž3, 3.4 .
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Proof. If r ) 2, then we may choose si , t i , and u i in distinct regular orbits of C G Ž wi .rC G ŽWi .. If r s 2, we just choose si and t i . If wi and w k are C G Ž w .-conjugate, then we may assume that si and sk are also C G Ž w .conjugate as well as t i and t k and also u i and u k . By Proposition 1, it follows that si is C G Ž w .-conjugate to no t k or u k ; and that no t i is C G Ž w .-conjugate to a u k . If x i , yi g si , t i , u i 4 , then x and y are in regular orbits of C C Ž w . where x s x 1 q ⭈⭈⭈ qx n and y s y 1 q ⭈⭈⭈ qyn , because each Wi is C-invariant and V is a faithful C-module. Furthermore, w and y are not conjugate under C C Ž w . unless x i s yi for all i. So C C Ž w . has at least r n regular orbits on V. If C G Ž w . : C, we are done. Letting LrC s CC G Ž w .rC ( C G Ž w .rC C Ž w ., we may assume that L ) C and k ) 1. We may assume that W1 , . . . , Wk 4 is a regular orbit of LrC and so C s NLŽW1 .. Let z s s1 q x 2 q ⭈⭈⭈ qx n with x i in a regular orbit of C G Ž wi .rC G ŽWi ., but such that x i is not C G Ž wi .-conjugate to si for 2 F i F k. From Proposition 1, it follows that C G Ž w . l C G Ž z . : NLŽW1 . s C and thus C G Ž w . l C G Ž z . s C C Ž w . l C C Ž z . s 1, with the last equality from the last paragraph. So z is in a regular orbit of C G Ž w .. Likewise, so is t 1 q x 2 q ⭈⭈⭈ qx n provided x i is in a regular orbit of C G Ž wi .rC G ŽWi . and x i is not C G Ž wi .-conjugate to t i for 2 F i F k. Thus there are at least r ny kq1 Ž r y 1. ky 1 distinct elements lying in regular orbits of C G Ž w . which lie in distinct orbits of C C Ž w .. Because
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A solvable primitive permutation group S on ⍀ has a unique minimal normal subgroup M. Furthermore, M transitively and regularly permutes the elements of ⍀ so that < ⍀ < s < M < is a prime power. For each ␣ in ⍀, we have that S s MS␣ with M l S␣ s 1 and that the actions of S␣ on M and ⍀ are permutation isomorphic. A group is supersolvable if every chief factor has prime order. If S is supersolvable, then < M < s < ⍀ < s p for a prime p and S is isomorphic to a subgroup of the semi-direct product Z p ( Z py1 that contains Z p . In particular, S␣ l S s 1 for distinct ␣ ,  g ⍀. COROLLARY 3. Suppose that V is a faithful G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the supersol¨ able factor group GrC. Suppose that wi g Wi and set w s w 1 q w 2 q ⭈⭈⭈ qwp . If C G Ž wi .rC G ŽWi . has at least r G 3 regular orbits on Wi for each i, then C G Ž w . has a regular orbit on V. Indeed, then C G Ž w . has at least r regular orbits on V except possibly when r s 3 s p and GrC ( S3 . Proof. By Lemma 2, we may assume that CC G Ž w .rC is not cyclic and thus p ) 2. Since GrC is isomorphic to a subgroup of Z p ( Z py1 , it follows that CC G Ž w .rC is transitive on the Wi . We may choose si , t i , and u i in distinct regular orbits of C G Ž wi .rC G ŽWi . and assume by Proposition 1 that no two of si , t j , or u k are ever C G Ž w .-conjugate. Now C G Ž w . l C G Ž s . for s s s1 q t 2 q u 3 q ⭈⭈⭈ qu p must stabilize both W1 and W2 and thus lie in C Žsee comments preceding this Corollary.. Thus C G Ž w . l C G Ž s . : C C Ž w . l C C Ž s . : li C C ŽWi . s 1. We have that s s s1 q t 2 q u 3 q ⭈⭈⭈ qu p lies in a regular orbit of C G Ž w ., as desired. In fact, this argument shows there exist Ž 3r . s r Ž r y 1.Ž r y 2.r6 elements of V which lie in distinct regular orbits of C G Ž w .. For r ) 3, C G Ž w . has at least r regular orbits on V. To complete the proof, we need just show that C G Ž w . has at least 3 regular orbits on V when r s 3 and 3 - p. But then we have that s, s1 q u 2 q t 3 q ⭈⭈⭈ qt p and t 1 q u 2 q s3 q ⭈⭈⭈ qs p lie in three distinct regular orbits of C G Ž w .. Gluck’s Permutation Lemma wMW, Theorem 5.6x characterizes those solvable primitive permutation groups Ž S, ⍀ . for which S does not have a regular orbit on the power set of ⍀. In all these cases, < ⍀ < - 10. An important case in the proof is where < ⍀ < is prime, i.e., S is supersolvable. We quote this. LEMMA 4 ŽGluck.. If S is a supersol¨ able primiti¨ e permutation group on ⍀, then there is a subset ⌬ : ⍀ such that Stab S Ž ⌬ . s 1 unless < ⍀ < s 3, 5, or 7.
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DEFINITION. If V is a G-module, we say that ¨ g V lies in a semi-regular G-orbit if C G Ž ¨ . has a regular orbit on V. Furthermore, we say that ¨ is in a type k semi-regular orbit if C G Ž ¨ . has at least k regular orbits on V. LEMMA 5. Suppose that V is a faithful G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the supersol¨ able factor group GrC. Suppose that w 1 , x 1 g W1 lie in distinct NG ŽW1 .rC G ŽW1 .-semi-regular orbits of types r and s. If rs ) 2 or if rs s 2 s p, then there exist u and ¨ in distinct semi-regular G-orbits of V of types and with G rs. Proof. Choose wi , x i g Wi that are G-conjugate to w 1 and x 1 , respectively. Now each wi Žrespectively, x i . is a type r Žresp. type s . semi-regular orbit of NG ŽWi .rC G ŽWi .. Also no wi is G-conjugate to an x j by Proposition 1 applied to C G Ž0.. We may assume that GrC / 1 and thus that p is prime. Assume first that p f 3, 5, 74 so that we may apply Gluck’s lemma and choose Žwithout loss of generality. m with 1 F m - p such that StabG r C 1, . . . , m4 s 1. If u s w 1 q ⭈⭈⭈ wm q x mq1 q ⭈⭈⭈ qx p , then C G Ž u. : C and C G Ž u. s C C Ž u. has at least r m s pym G rs regular orbits on V. Also C G Ž ¨ . has at least s m r pym G rs regular orbits on V where ¨ s x 1 q ⭈⭈⭈ x m q wmy1 q ⭈⭈⭈ qwp . Furthermore u and ¨ are not conjugate in G if p is odd. If p s 2, then w s w 1 q w 2 is in a semi-regular orbit of G by Lemma 2 and w is not conjugate to u by Proposition 1. Thus, we may now assume that p s 3, 5, or 7. By the hypotheses, rs ) 2. We may assume that r G s, and so r ) 2 or r s 2 s s. Now let u* s w 1 q ⭈⭈⭈ qwpy1 q x p and ¨ * s w 1 q ⭈⭈⭈ qwpy2 q x py1 q x p , so that u* and ¨ * are not G-conjugate. Now C G Ž u*.CrC and C G Ž ¨ *.CrC are both cyclic. If r s s s 2, then Lemma 2 shows that C G Ž u*. and C G Ž ¨ *. each have Žat least. two regular orbits on V. We thus assume that r ) 2. Observe that C G Ž u*. must stabilize Wp and CC G Ž u*.rC faithfully permutes W1 , . . . , Wpy14 . Lemma 2 shows that C G Ž u*.rC G Ž u*. l C G ŽW1 q ⭈⭈⭈ qWpy1 . has at least r regular orbits on W1 q ⭈⭈⭈ qWpy1. Since C G Ž u*. : NG ŽWp ., we have that C G Ž u*.rC G Ž u*. l C G ŽWp . has at least s regular orbits on Wp . Hence C G Ž u*. has at least rs regular orbits on V s W1 q ⭈⭈⭈ qWp . Also Corollary 3 shows that C G Ž w . has a regular orbit on V where w s w 1 q w 2 q ⭈⭈⭈ qwp . Because u* and w are not conjugate, the proof is complete. COROLLARY 6. Suppose that V is a faithful G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the supersol¨ able factor group GrC. Assume that C / 1 and that NG ŽW1 .rC G ŽW1 . has a regular orbit on W1. Then there exist u and ¨ in distinct semi-regular G-orbits of V of types and with G < W1 < ) 2.
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Proof. Since C / 1, indeed < W1 < ) 2. If w g W1 is in a regular orbit NG ŽW1 .rC G ŽW1 ., then w / 0 lies in a type < W1 < semi-regular of NG ŽW1 .rC G ŽW1 ., while 0 does lie in a different semi-regular orbit of NG ŽW1 .rC G ŽW1 .. Apply Lemma 5. Suppose that V is a vector space of dimension n over GF Ž q .. We may then label the elements of V by those of GF Ž q n . in a one-to-one fashion. We let ⌫ Ž V . or ⌫ Ž q n . denote the group of semi-linear transformations of V, i.e., ⌫ Ž V . s x ª ax <0 / a g GF Ž q n ., g GalŽ GF Ž q n .rGF Ž q ..4 . In particular, ⌫ Ž V . has a cyclic normal subgroup ⌫0 Ž V . of order q n y 1 that acts fixed-point-freely on V Žby multiplications. and the factor group ⌫ Ž V .r⌫0 Ž V . is cyclic of order n Žisomorphic to the Galois group.. In particular, ⌫ Ž V . is metacyclic. For a faithful G-module V, we will say G F ⌫ Ž V . if the elements of V may be labeled in a way that yields G a subgroup of ⌫ Ž V .. The semi-linear groups ⌫ Ž V . are supersolvable and play an important role in representation theory. We gather some known facts about supersolvable groups, the last of which characterizes primitive supersolvable linear groups as semi-linear groups. LEMMA 7.
Let G be supersol¨ able. Then
Ža. GrFŽ G . is abelian; Žb. G has a normal Sylow-p-subgroup if p is the largest prime di¨ isor of < G <; and Žc. If V is a faithful primiti¨ e and finite G-module, then G F ⌫ Ž V .. Proof. Parts Ža. and Žb. are immediate from the definition of supersolvability and are well known. Let A be a maximal abelian normal subgroup of G. Again, it is well known and direct from the definition of supersolvable that A s C G Ž A.. Since V is a primitive G-module and A is normal in G, VA is homogeneous by Clifford’s Theorem. Because VA is homogeneous and A s C G Ž A., it follows that G F ⌫ Ž V . Že.g., see wMW, Lemma 2.2x.. Recall, for positive integers q ) 1 and n, a prime divisor r of q n y 1 is called a Zsigmondy prime di¨ isor of q n y 1 if r does not divide q m y 1 whenever 0 - m - n. In this case, n is the order of q Žmod r . and thus n divides Ž r . s r y 1 for Euler’s -function. In particular, gcdŽ r, n. s 1. The Zsigmondy Prime Theorem asserts that q n y 1 always has a Zsigmondy prime divisor except when q n s 2 6 or when n s 2 and q is a Mersenne prime. LEMMA 8. Suppose that V is a faithful FG-module with G supersol¨ able and that G acts transiti¨ ely on V y 04 . Set < V < s q n where q s < F <. Ži .
Then G : ⌫ Ž V .; and
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Žii. If r is a Zsigmondy prime di¨ isor of q n y 1 and R g Syl r Ž G ., then R : ⌫0 Ž V . l G s FŽ G . s C G Ž R .; Žiii. If q n s 2 6 , then FŽ G . s G l ⌫0 Ž V . is cyclic of order 21 or 63. Proof. The transitivity hypothesis implies that V has no proper G-modules and is thus irreducible. Suppose that V s V1 [ ⭈⭈⭈ [ Vn for modules Vi that are permuted by G. If ¨ i g Vi is non-zero for each i, then ¨ 1 cannot be G-conjugate to ¨ 1 q ¨ 2 . Thus the transitivity hypothesis implies that n s 1 and that V is a primitive G-module. By Lemma 7Žc., G : ⌫ Ž V .. If r is a Zsigmondy prime divisor of q n y 1 and R g Syl r Ž G ., then 1 / R by transitivity and also R : ⌫0 Ž V . because gcdŽ r, n. s 1 Žsee paragraph preceding this lemma.. Statements of Žii. and Žiii. are immediate from Lemmas 6.4 and 6.5Žb., respectively, of wMWx. PROPOSITION 9. If G s ⌫ Ž V . and 0 / x g V, then C G Ž x . has at least three non-zero regular orbits on V unless < V < s 2 n with n F 3 or < V < s 3. Proof. Set < V < s q n for a prime power q. If n s 1, then C G Ž x . s 1 and the result is trivial. Now G acts transitively on V 噛 and so we may replace x by any non-zero element of V. If is a field automorphism of order n, then ² : is the centralizer of an element of V and so we may assume that C G Ž x . s ² :. Now C s CŽ x . is cyclic of order n. If n is prime, then C V Ž C . has order q and every element in V y C V Ž C . is in a regular orbit of C. Since Ž q n y q .rn G 3 unless q s 2 and n F 3, the result follows when n is prime. If n is a power of two, then C has a unique involution t. Every element of V y C V Ž t . is in a regular orbit of C. Since
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Suppose that V is faithful, irreducible G-module and that VC s W1 [ ⭈⭈⭈ [ Wn for C-invariant submodules Wi that are permuted faithfully by the factor group GrC with n ) 1. Then C / 1, since otherwise C V Ž G . / 1. Also W1 is an irreducible H-module where H s NG ŽW1 .rC G ŽW1 ., because G is isomorphic as a linear group to an irreducible subgroup of the wreath product H wrŽ GrC . Žsee wMW, Lemma 2.8x.. Because C / 1, also H / 1 and < W1 < ) 2. THEOREM 10. Suppose that V is a faithful, irreducible G-module and that VC s W1 [ W2 [ ⭈⭈⭈ [ Wp for C-in¨ ariant submodules Wi that are permuted faithfully and primiti¨ ely by the factor group GrC / 1. If G is supersol¨ able and NG ŽW1 .rC G ŽW1 . is isomorphic to a subgroup of ⌫ ŽW1 ., then G has two distinct semi-regular orbits on V of types and with ) 2 except possibly when: Ži. p s 2, < W1 < s 2 2 , and s 2; Žii. p s 3, < W1 < s 2 2 , GrC ( S3 , and there exist x and y in V in distinct semi-regular orbits of G with y in a type 2 semi-regular orbit and such that C G Ž x . has three distinct orbits in V that lie in regular orbits of C FŽG.Ž x .; or Žiii. p s 2, < W1 < s q 2 for a Mersenne prime q, and G has a type 3 semi-regular orbit. Proof. Because the primitive permutation group GrC / 1 is supersolvable, p is prime and GrC is isomorphic to a subgroup of Z p ( Z py1. Set Hi s NG ŽWi . and set Ci s C G ŽWi . and < Wi < s q n for a prime power q s < F < where F is the underlying field. Because V is an irreducible G-module, also W1 is an irreducible H1rC1-module and < W1 < / 2 Žsee comments before the theorem.. If H1rC1 has a regular orbit on W1 , then G satisfies the principle conclusion of the theorem by Corollary 6. Thus we may assume that H1rC1 is not isomorphic to a subgroup of ⌫0 ŽW1 . and hence that n ) 1. If H1rC1 has at least two type 2 semi-regular orbits on W1 , then Lemma 5 shows that G has two distinct semi-regular orbits on V of types and with ) 2. But, by Lemma 7, every non-zero element of W1 is in a type 2 semi-regular orbit of H1rC1 except possibly when < W1 < s 4 and H1rC1 ( ⌫ Ž2 2 . ( S3 . Hence, in all cases, we may assume that H1rC1 acts transitively on the non-zero vectors of W1. We denote by Di that subgroup of Hi defined by DirCi s ⌫0 ŽWi . l HirCi . Then DirCi acts on Wi by field multiplications and Ci s C D i Ž x . for all non-zero x in Wi . Now, if < Wi < s q n for a prime power q, then DirCi and HirDi are cyclic with < DirCi < dividing q n y 1 and < HirDi < dividing n. Set D s Fis1 to p Di . Because Fis1 to p Ci s 1 and Fis1 to p Hi s C, it follows that D and CrD are abelian with exponents dividing q n y 1 and n Žrespectively.. Furthermore, if x i is a non-zero element of Wi , then C D Ž x 1 q ⭈⭈⭈ qx p . : lC D i Ž x i . : lCi s 1. Unless q n s q 2 for a
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Mersenne prime q, then DirCi s FŽ HirDi . by Lemma 8 and the Zsigmondy prime theorem. Standard arguments Žsimilar to those of wMW, Proposition 9.5Ža.x. then show that D s FŽ C . unless q n s q 2 for a Mersenne prime q. The primitive permutation group GrC has a unique minimal normal subgroup MrC of order p. We claim that CrD is central in MrD. If FŽ G . s FŽ C . s D, then GrD is abelian by Lemma 7. If FŽ G . ) FŽ C . s D, then MrD s CrD = FŽ G .rD. If p ) n, then p is the largest prime divisor of < MrD <, whence MrD has a normal Sylow-p-subgroup PrD by Proposition 9 and so MrD s CrD = PrD. The claim holds unless FŽ C . ) D and p F n. By the last paragraph, this can only happen when q is a Mersenne prime and n s 2 s p. But, in this exceptional case, G has a type 3 semi-regular orbit by Proposition 9 and Lemma 2 and thus exception Žiii. of this theorem holds. Thus we may assume that CrD is central in MrD. Now MrC transitively permutes the subgroups C l DirD, while CrD is central in MrD. Thus C l D 1 s ⭈⭈⭈ s C l Dp . But lDi s D and so D s C l D 1 s ⭈⭈⭈ s C l Dp . Now CrD is isomorphic to a subgroup of H1rD 1 and is hence cyclic. If x i g Wi is non-zero for all i, note that C D Ž x i . : Ci and thus Fi C D Ž x i . s 1. If u s u1 q ⭈⭈⭈ qu p and ¨ s ¨ 1 q ⭈⭈⭈ q¨ p with u i , ¨ i g Wi , then we claim that C C Ž u. l C C Ž ¨ . s 1 provided that u i or ¨ i is non-zero for each i and that ¨ j , for some j, is in a regular orbit of C G Ž u j .rC j . The claim is valid because C D Ž u. l C D Ž ¨ . s 1 and C C Ž u. l C C Ž ¨ . : C C Ž u j . l C C Ž ¨ j . : C j l C : Dj l C s D. If x i is a non-zero element of Wi , we let r be the number of regular orbits and s be the number of non-zero orbits of C G Ž x i .rCi on Wi . These values are independent of choices because H1rC1 acts transitively on the non-zero vectors of W1 and GrC transitively permutes the Wi . Since H1rC1 does not have a regular orbit on W1 , s ) r G 1 with the last inequality by Proposition 9. Fix non-zero x i g Wi and set y s x 1 q ⭈⭈⭈ qx py1. If z s z1 q ⭈⭈⭈ qz p with z i g Wi with z p / 0 and z py1 in a regular orbit of C G Ž x py1 .rC py1 , then z is in a regular orbit of C C Ž y . by the next to last paragraph. If, in addition, z py 1 is not conjugate to z i for all i - p y 1, then C G Ž y . l C G Ž z . must normalize Wpy 1 and Wp and thus lies in C. With this additional condition, we have that z is in a regular orbit of C G Ž y .. Furthermore a s a1 q ⭈⭈⭈ qa p with a i g Wi can only be C C Ž y .-conjugate to z if a i is C G Ž x i .rCi-conjugate to z i for i - p. Thus we have rs py 2 elements of V that lie in regular orbits of C G Ž y . and in distinct orbits of C C Ž y .. Since
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one of z i for i - 3 is in regular orbit of C G Ž x i .rCi . In this case, we have 2 ⭈ 1 ⭈ 2r2 regular orbits of C G Ž y .. Summarizing, C G Ž y . has 2 regular orbits on V except possibly when r s 1 and p s 2. When r s 1, then Proposition 9 shows that < W1 < s 2 2 and H1rC1 ( ⌫ Ž2 2 . ( S3 . Next we let x s x 1 q ⭈⭈⭈ qx p . If z s z1 q ⭈⭈⭈ qz p with z i g Wi with some z i in a regular orbit of C G Ž x i .rCi , then z is in a regular orbit of C C Ž x . by above because every x i is non-zero. Let u i g Wi be in a regular orbit of C G Ž x i .rCi , so that u i / 0 because HirCi has no regular orbit on Wi . Then u s 0 q ⭈⭈⭈ q0 q x py1 q u p and ¨ s u1 q ⭈⭈⭈ qu py2 q x py1 q 0 lie in regular orbits of C G Ž x .. If p ) 3, then u and ¨ cannot even be G-conjugate and C G Ž x . has at least two regular orbits on V. Also C G Ž x . has at least r regular orbits on V, because u and 0 q ⭈⭈⭈ q0 q x py 1 q t p can only be C G Ž x .-conjugate if t p and u p are C G Ž x p .rC p-conjugate. If GrC is cyclic, then u and 0 q ⭈⭈⭈ q0 q u p are in distinct regular orbits of C G Ž x . on V. Hence C G Ž x . has two regular orbits on V provided p ) 3, GrC is cyclic or r ) 1; i.e., x is a type 2 semi-regular orbit unless p s 3, GrC ( S3 , < W1 < s 2 2 , and H1rC1 ( ⌫ Ž2 2 . ( S3 . But do observe in this case that we do have one regular orbit of C G Ž x . on V. Observe, in this exceptional case, that the three elements 0 q x 2 q u 3 , 0 q u 2 q u 3 , and 0 q 0 q u 3 lie in distinct orbits of C G Ž x . and are in regular orbits of C FŽ G.Ž x ., because FŽ G .rD has order three and transitively permutes the Wi and C D Ž x . s 1. Now x and y cannot be G-conjugate because all the x i are non-zero. Both are in semi-regular orbits of G and at least one lies in a type 2 semi-regular orbit. Indeed both lie in type 2 semi-regular orbits of G or exception Ži. or Žii. applies. This completes the proof. We used the supersolvability of G in critical ways in Theorem 10 Žsee Example 13.. This will be used again in the next two results. LEMMA 11. Suppose that V is a faithful G-module and V s V1 [ ⭈⭈⭈ [ Vp for subspaces that are permuted primiti¨ ely by G. Suppose that Hi s NG Ž Vi . and that V1 can be written as a direct sum of q ) 1 subspaces that are permuted primiti¨ ely by H1. If p ) q and G is supersol¨ able, then V can be written as a direct sum of q ) 1 subspaces that are permuted primiti¨ ely by G. Proof. Let C be the kernel of the permutation action of G on V1 , . . . , Vp 4 . We may write V1 as a direct sum W11 [ ⭈⭈⭈ [ W1 q of subspaces primitively and faithfully permuted by H1rD 1 for a normal subgroup D 1 of H1. For g, h in the same right coset of H1 in G, observe that W11 , . . . , W1 q 4 g s W11 , . . . , W1 q 4 h. Hence, we may write Vi as a direct sum Wi1 [ ⭈⭈⭈ [ Wi q of subspaces primitively and faithfully permuted by HirDi for a normal subgroup Di of Hi that is G-conjugate to D 1. Furthermore, if we set D s lDi : lHi s C, then the group GrD permutes the set Wi j 4 of pq subspaces transitively and faithfully.
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Since GrC is supersolvable, GrC has a unique minimal normal subgroup MrC of prime order p that transitively permutes the Vi and a cyclic factor group GrM whose order divides p y 1. Now HirDi has a unique minimal normal subgroup of prime order q and a cyclic factor group whose order divides q y 1. Since lHi s C and lDi s D, it follows from the hypothesis p ) q that p must be larger than all prime divisors of < CrD 5 GrM <. By the supersolvability of G, GrD has a normal Sylow-psubgroup PrC of order p. Now PrC acts on Wi j 4 with q faithful orbits and we may write V s X 1 [ ⭈⭈⭈ [ X q for P-invariant submodules X k that are permuted transitively by GrP. If LrP is the kernel of the action of GrP on X 1 , . . . , X q 4 , then GrL even acts primitively on X 1 , . . . , X q 4 because q is prime. THEOREM 12. Suppose that V is a faithful, irreducible G-module for a supersol¨ able group G. Assume that V is an imprimiti¨ e G-module and choose p ) 1 as small as possible so that V may be written V s W1 [ ⭈⭈⭈ [ Wp for subspaces Wi that are permuted primiti¨ ely by G. If W1 is imprimiti¨ e as an NG ŽW1 .rC G ŽW1 .-module, then Ži. p s 2 and G has a type 3 semi-regular orbit; or Žii. G has two distinct semi-regular orbits of types and on V such that G min 3, p4 . Proof. Now W1 is imprimitive as an H-module where H s NG ŽW1 .rC G ŽW1 .. Choose q ) 1 as small as possible such that W1 may be written W1 s U1 [ ⭈⭈⭈ [ Uq as a direct sum of q subspaces that are permuted primitively by HrD for a normal subgroup D of H. By Lemma 11 and choice of p, it follows that q G p. First suppose that U1 is imprimitive as a NH ŽU1 .rC H ŽU1 .-module. Arguing by induction on dimŽ V ., we may conclude that: Ža. H has two distinct semi-regular orbits of types r and s on W such that rs G min 3, q4 ; or Žb. q s 2 and H has a type 3 semi-regular orbit on W. On the other hand, if U1 is a primitive NH ŽU1 .rC H ŽU1 .-module, then Lemma 7 shows NH ŽU1 .rC H ŽU1 . is isomorphic to a subgroup of ⌫ ŽU1 .. Then we apply Theorem 10 to conclude that H and W satisfy Ža. or Žb. above or that: Žc. q s 3, < U1 < s 2 2 , H is a 2, 34 -group, and there exist x and y in W1 in distinct semi-regular orbits of H with y in a type 2 semi-regular orbit and such that C H Ž x . has three distinct orbits in V that lie in regular orbits of regular C FŽ H .Ž x .. Whether U1 is primitive or imprimitive, H and W satisfy Ža., Žb., or Žc..
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If q s 2, then also p s 2 by Lemma 11 and choice of p. If H and W satisfy Žb., then p s 2 and G has a type 3 semi-regular orbit by Lemma 2. Conclusion Ži. of the theorem is met here. If H satisfies Ža., then Lemma 5 shows that G has two distinct semi-regular orbits of types and on V such that G min 3, q4 . Conclusion Žii. follows here as q G p. Similarly, Lemma 5 shows that Conclusion Žii. holds if H satisfies Žc. and p s 2. To complete the proof, we may assume that Žc. above holds and p s 3. Recalling V s W1 [ W2 [ W3 , we let Hi s NG ŽWi ., let Ci s C G ŽWi ., and C s lHi so that H s H1rC1 and lCi s 1. Let FirCi s FŽ HirCi . and observe that lFi s FŽ C .. Now GrC primitively and faithfully permutes W1 , W2 , W3 4 and is isomorphic to a transitive subgroup of S3 . Because G is a supersolvable 2, 34 -group, G has a normal Sylow-3-subgroup. But charŽ V . s 2 and V is irreducible, so that O 2 Ž G . s 1 and FŽ G . is the Sylow-3-subgroup of G. Now FŽ G .rFŽ C . is isomorphic to the Sylow-3-subgroup MrC of GrC and transitively permutes W1 , W2 , W3 4 . Also CrFŽ C . is the 2-group and so MrFŽ C . s CrFŽ C . = FŽ G .rFŽ C .. Since FŽ G .rFŽ C . transitively permutes the groups Fi l CrFŽ C . and centralizes CrFŽ C ., it follows that F1 l C s F2 l C s F3 l C s FŽ C .. By Žc. above, we may choose x i , yi g Wi in distinct semi-regular orbits in Wi of HirCi of types 1 and 2 Žrespectively. and such that C G Ž x i .rCi : HirCi has three distinct orbits on Wi that lie in regular orbits of C G Ž x i . l FirCi . We may assume that no x i and y k are G-conjugate. Suppose that si , t i g Wi lie in distinct regular orbits of C G Ž yi .rCi for each i. Also choose a i , bi , c i g Wi in distinct orbits of C G Ž x i . : Hi , each of which is in a regular orbit C G Ž x i . l FirCi . For i / j, we may assume that a i and bj are not conjugate via C G Ž x i q x j ., etc. Then C G Ž y 1 q x 2 q x 3 . l C G Ž s1 q b 2 q c 3 . : lHi s C. Now C C Ž y 1 q x 2 q x 3 . l C C Ž s1 q b 2 q c 3 . : C C Ž y 1 . l C C Ž s1 . : C1 l C : F1 l C s FŽ C .. Because FŽ C . : Fi , it follows that C FŽ C .Ž y 1 q x 2 q x 3 . l C FŽC .Ž s1 q b 2 q c 3 . : lCi s 1. Hence C G Ž y 1 q x 2 q x 3 . l C G Ž s1 q b 2 q c 3 . s 1 and so s1 q b 2 q c 3 is in a regular orbit of C G Ž y 1 q x 2 q x 3 .. Likewise, t1 q b 2 q c3 and s1 q a2 q c3 are in regular orbits of C G Ž y 1 q x 2 q x 3 .. Thus y 1 q x 2 q x 3 lies in a type 3 semi-regular orbit of G Žsee Proposition 1.. Now similar arguments show that s1 q t 2 q a3 , s1 q t 2 q b 3 , and s1 q t 2 q c 3 lie in regular orbits of C G Ž y 1 q y 2 q x 3 . and these lie in distinct orbits of C G Ž y 1 q y 2 q x 3 . by Proposition 1. Both y 1 q y 2 q x 3 and y 1 q x 2 q x 3 lie in a type 3 semi-regular orbits of G. Because y 1 q x 2 q x 3 and y 1 q y 2 q x 3 are not G-conjugate, conclusion Žii. of this theorem is met. Proof of Theorem A. Arguing by induction on dimŽ V ., we may assume that V is irreducible. If V is a primitive G-module, then G : ⌫ Ž V . by Lemma 7, whence Proposition 9 gives the existence of a semi-regular orbit.
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Otherwise we may write V s W1 [ ⭈⭈⭈ [ Wp for subspaces Wi that are permuted primitively by G and do so with p as small as possible. If W1 is imprimitive as a NG ŽW1 .rC G ŽW1 .-module, then Theorem 12 gives the existence of a semi-regular orbit. If W1 is primitive as a NG ŽW1 .rC G ŽW1 .module, then Theorem 10 gives the existence of a semi-regular orbit. We next give an example where Theorem A fails for a group G that is the wreath product H wr S of supersolvable groups acting on V s W G , but where the conclusion of Theorem A is valid for the action of H on W. While the action here is not coprime, Isaacs’ question of an orbit as large as < G < 1r2 also fails here. EXAMPLE 13. Let H s S3 s S, so that H acts irreducibly on a vector space W of order 2 2 and has exactly one semi-regular orbit. The wreath product G s H wr S of order 2 4 3 4 acts irreducibly on a vector space V of order 2 6 . The G-orbits in V have size 1, 9, 27, and 27 and so no G-orbit has size as large as < G < 1r2 . Furthermore, for each ¨ g V,
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I. M. Isaacs, Large orbits in nilpotent actions, Proc. Amer. Math. Soc., in press. O. Manz and T. Wolf, ‘‘Representations of Solvable Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1993. P. P. Palfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77 Ž1982., 127᎐137. D. Passman, Groups with normal Hall-p⬘-subgroups, Trans. Amer. Math. Soc. 123 Ž1966., 99᎐111. G. Robinson, Further reductions for the k Ž GV .-problem, J. Algebra 195 Ž1997., 141᎐150. A. Turull, Supersolvable automorphism groups of solvable groups, Math. Z. 183 Ž1983., 47᎐73.