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Periodic groups in which the centralizer of an involution has bounded order
JOI‘RNAL
OF ALGEBRA
Periodic
64,
Groups
285-291
(1980)
in which the Centralizer Bounded Order
of an Involution
Has
B. HARTLEY Department
of Mathematics,
of Manrhcster,
University
Manchester,
En&and
AND ‘RI. Math.
Institut
MEIXNER
der Universitci’t,
Communicated Received
1.
6300
Giessetl,
Germarty
by B. Huppert May
4, I979
INTRODUCTION
The properties of involutions in groups, and in particular the centralizers of involutions, often exert a very strong influence over the structure of the group. One of the best known illustrations of this is the celebrated theorem of Brauer and Fowler [2], which asserts that there are only finitely many finite simple groups in which the centralizer of an involution has given order. This has been refined by Fong [3] as follows: 1 (Fong). There exists a function fi(c) such that ij G is a finite containing an involution i such that 1 C,(i)’ < c, then G : O,,,(S(G))/ .<
THEOREM
group fd4
Here S(G) denotes the soluble p-nilpotent subgroup of G. On the other hand, for infinite remarkable result:
radical groups,
of G and O,,,(G) Sunkov
the largest
normal
[6] has proved the following
TIIICOREV 2 (Sunkov). Let G be a periodic group containing an involution i with Jinite centralizer. Then G contains a soluble normal subgroup of jnite index, and so is locally finite.
In this paper we shall prove a theorem though both are used in its proof.
which
extends both of these results,
285 0021-8693!‘80/050285-07%02.00,!0 Copyright s. 1380 by Academic Press, Inc. All rights of reproduction in any form rescrvcd.
286
HARTLEY
Ah-D
YEIXNER
THEOREM. There exists a junction j(m) such that if G is a periodic group containing an involution i with 1C,(i)1 = m, then G contains a nilpotent subgroup of class at most two and index ,< j(m).
We make no attempt to describe such a function jexplicitly, except in certain special cases. By .%rnkov’s Theorem, G is locally finite, and a routine inverse limit argument, applied to the set of finite subgroups of G containing i (see [5, Chap. 1, Sect. K]) allows us to reduce the proof of the theorem to the case when G is finite. Thus this paper is really about finite groups. By Fong’s Theorem, G then contains a soluble normal subgroup of bounded index, and so we can assume G is soluble. We could even assume G 2-nilpotent, but this will not be necessary. In proving the theorem, it will usually be more convenient to think of a finite soluble group G admitting an involutory automorphism (z such that ! Co(o~)( = m. We have to show that G has a nilpotent subgroup of class at most two and bounded index. We do this in two stages. First we prove the result when G is nilpotcnt, and then we show that the Fitting subgroup F(G) of G has bounded index. The second part of the argument can be carried out when two is replaced by an arbitrary prime and will be dealt with in that generality in a later publication, but because of the simplicity of the argument for the prime two, it seems worth giving it separately. The following fact will be useful: LEMMA 1.1. Let G be a finite group admitting an automorphism (Y, and let N be a normal a-invariant subgroup of G. Then 0) (ii)
I Gd4I Ij(i(or)l,
G I Cd4 I G 1) = 1, then CGIN(a) = Cc(~)N/N.
Proof. (i) The proof of this can be found for example in [1], but we give it for convenience. If K/N = CcIN(ti), then by replacing G by K we can obviously assume that OLoperates trivially on G/N. Then the map 8: g --f [g, a] = g-igu maps G into N, and so 1im 8 1 < / N I. If g, , g, E G, then O(gi) = O(ga) if and only if gag,’ E C’,(z) =: C. Hence I im 0 1 = 1G : C j = : G I/i C I < 1N 1, and so j G : N ) = I G I/I N ! < I C /, as required. (ii)
This is well known,
see for example [4, 1.18.6J
Lemma 1.l in fact extends immediately to the locally finite case. 2.
NILPOTENT
GROUPS
LEMMA 2.1. Let G be afinitep-group ojclass 3 and suppose that 1 G’/y&G)/ p”. Then G contains a subgroup U such that
-
CENTRALIZERS
(i)
OF
287
INVOLUTIONS
C: > G’,
(ii)
1 G : U 1
(iii)
CT is nilpotent of class ai mosttwo.
We use {r,(G)}
to denote the lower central series of a group G, and G’ y--.y*(G).
Proof. Let Z(G) be the centre of G and Zi :- G’ n Z(G) 3 ya(G). Thus IG’:Z,I..., x,)). Let II = <:.v,,..., s,). Since I(G/Z,)’ / .< p”, each element of G/Z, has at most p71 conjugates and so, if U/Z, = C,,,,(NZJZ,), then jG: Uj G’. Finally, [H, li, U] < [Z, , C’] =- 1, and by the three subgroups lemma, [H, c”] = 1. Hence c” .< C,*(H) =- Z, , and so U is nilpotent of class at most two. It will be convenient to say that an automorphism every element of that group.
inverts a group,
if it inverts
LEMMA 2.2. Let G beafinite p-group admitting an involutory automorphism LY. IfG L--:[G, a], then a:inverts ~,(G)/Y+~(G) if i isodd, and centralizes~(G)/Y~+~(G) if i is even.
Proof. We may assume G #- 1, in which we see [Xl ,..., hence [Xl ,..., The p-group.
case p + 2. Since G -=: [G, a], that 01inverts G/y,(G). If xi ,..., xi E G, then the coset of the commutator xi] modulo yiT1(G) depends only on the cosets of xi ,..., Xi mod G’ and mod yi+i(G) we have [xi ,..., xi]” 7-1 [xi” ,..., .qa] :+ [XL’ ,..., A-;‘] F xi]t-I)‘, from which the assertion follows. next lemma deals with the special case of the theorem when G is a When p + 2, notice that we obtain an explicit bound.
I,EMMA 2.3. Let G be a jnite p-group admitting an involutory automorphism cywith / Co(o)1= p”l. Then
(i) If p # 2, G contains an a-invariant subgroupof nilpotency classat mosttwo and index at mostpfW, wheref2(m) == la $ 22 -j- .** + m2. (ii) If p == 2, then G containsan o-invariant abelian subgroupof index at most2fJnr), wheref3(m) is a certain integer-valuedfunction.
Proof. (i) We use induction on m. If m = 0 then a! operates fixed point freely on G, and so G is abelian. Thus we may put f*(O) = 0. Now suppose that m > 0 and the statement has been proved for smaller values. If G > [G, a], then by Lemma 1.l(ii) we have I G : [G, a]1 := pi, where 0 < i < m, and for G, = [G, a] we have ] Ccl(~)1 = P”‘--~. By induction, G, contains an or-invariant subgroup of nilpotency class at most two and index at most pfkJm .
288
HARTLEY
AND
MEIXNER
Kow suppose that G [G, a]. Supp osc that G has nilpotency class t, and put s = t if t is odd, or s =I t J- 1 if t is even. Then, by Lemma 2.2, a inverts ys(G). We may suppose that s :l 1, and hence WC may choose an element .v E rs-i(G)\rz(G) such that xv E rJ(G). By Lemma 2.2, (Y centralizes subgroup of G. r,-l(G)lrs(G), and so H ---= is a normal a-invariant By Lemma 1.l(ii), I C,,,(a)1 = p’“-’ and so by induction, G/H contains an a-invariant subgroup U/H such that j G : lY < fle()“- li and ys( U) < H. Then yQ(U) .< rs(G). Let U, = [U, a] -= [c’, a, a] [4,111.13.3]. Then by Lemma l.l(ii), 1 U : C’, / = p”l j, for some j with 0 tii , and 1 U, : I’ 1 < pp. Since Q inverts U,/U; , 1’ is a-invariant. Also
Finally, f.(m - 1) + m : j(j - 1) < fa(m - 1) + m + m(m f2(m - 1) 7 m* - fa(m), and so 1’ is the subgroup sought.
1) =
(ii) Here G is a 2-group, and by forming the semidirect product G(a), we can suppose that OLis an involution in G. Let A be a maximal abelian normal subgroup of G. If A, = Q,(A) = {a E A : a2 = I}, then thinking of A, as a vector space over the field of two elements and (Y as a linear transformation - 1) of A, > we have (a - I)* = (Y? - 2cr $- 1 = 0. Hence the map a+a(a. determines an embedding of AliCA1(a) into C,L((Y), and we have 1 A, 1 < Pm. Therefore A has rank at most 2~2, that is, .4 is a direct product of at most 2m cyclic groups. By Lemma 1.1(i), 1 A : [A, a]1 .< 2”‘, and so [A, a] 3 B = {Us”: a E A}. If w E A, then clearly cy inverts W-W --: [w, x]. Hence (Y inverts B. Since the inverting automorphism lies in the centre of the automorphism group of every abelian group, we find that [G, LY] centralizes B. By Lemma 1.1, C,(B), which contains [G, or], has index at most 2” in G. Since A/B has bounded rank and bounded exponent, its order is bounded in terms of m, and hence its centralizer in G has bounded index. Thus G contains a subgroup U of bounded index which centralizes both A/B and B. Hence U/C,(A) is isomorphic to a subgroup of Hom(A/B, B). This is because, if II E U, the map T,,: a + [u, u] is a homomorphism of A into B which induces one of A/B into B, and the map u--f 7” is the desired embedding. Now A/B is a direct product of at most 2m cyclic groups of orders dividing 2”‘, and B has rank at most 2m, and hence ) Hom(AIB, B)I = 1Hom(A/B, Q,,,(B))1 is at most 2.iff13.Hence I U : C,(A)1 is bounded by a function of m. Since A is a maximal abelian normal subgroup of G, C,(.4) 3.; 4, and hence I G : .4 is bounded in terms of m.
CENTRALIZERS
289
OF INVOLUTIONS
COROLLARY 2.4. Let G be a finite nilpotent group admitting an involutory automorphismLX,and let ( Co(a)1 : = m. Then G has an cu-invariant subgroupof nilpotency classat mosttwo and index boundedby a function of m.
Proof. We note that if p is any prime not dividing m, then (Y operates fixed point freely on the Sylow p-subgroup G, of G, and so G, is abelian. I,emma 2.3 then gives the result. 1Ve should remark for completeness that in the situation of the corollary, there need not be an abelian subgroup of bounded index. This is because an cy of extraspecial group P of order p2”i-l (p #: 2) admits an automorphism order two which inverts the Frattini factor group; thus 1 C,(a)1 =p, while k may be arbitrarily large.
3. GENERAL We need the following
simple
CASF
representation-theoretic
lemma:
LEMMA 3.1. Let G = A(u) be the semidirectproduct of a $nite abelian normal subgroupA by a cyclic group (u) of prime order q. Let k be a jeld and V a kc-module, ofjnite dimensionover k, suchthat I’ : [I’, [A, u]]. Thendim, V r q dim, C,,(u). Proof. If k is any extension field of k and Y = v Ok k, then C,(u) :.C,(u) @k k. This well known fact is proved by writing v = $&o V @ w, where a is a k-basis for k Hence we may assume that k is algebraically closed. Let p :? 0 be the characteristic of k, and write A = P x Q, where P is the Sylow p-subgroup of .4 and Q the Sylow p’-subgroup; put P =: 1 and Q 1 A if p 0. We claim that I’ = [V, [Q, u]]. For [I’, [Q, u]] is a KG-submodule of I’. If it is proper, then it is contained in a maximal submodule W of I/. The irreducible module I,‘/ W is trivial for P and for [Q, a] and hence for [A, u] = [P, u] x [Q, u]. Hence [V, [A, u]] < W, contrary to the hypothesis. Thus we indeed have I - = [I;, [Q, u]], and by considering Q(u> instead of G, we may assume that A is a p’-group. Then I’, is completely reducible, and W [II’, [,4, u]] for any kA-submodule W of I,-. Sow cvcry irreducible kG-module X such that X = [X, [A, u]] is induced from .4 and so has dimension q. For if X, is an irreducible A-submodule of X, then the stabilizer of X, is .-I, since otherwise X,, is homogenous, A acts on it by scalar multiplication and [.4, u] is trivial on X. Clifford’s Theorem then shows that X is induced from A, as claimed. It follows from this that if T is any irreducible k-4-submodule of r, so that dim, T = 1, then ?’ 3 Tu (+ ... 5; Tu’ ’ is an irreducible KG-submodule. Therefore I’ is a sum of irreducihlc KG-modules, each induced from A, and must be the direct sum of a collection of them. Each of thcsc contributes one to dim,: C,(U), and the result follows.
290
HARTLEY
AND
MEIXNER
Proof of theorem. By the remarks following the statement of the theorem, we need to consider a finite soluble group G admitting an involutory automorphism (Ysuch that ! C,(a)1 = m, and have to show that G contains a nilpotent subgroup of classat most two and index bounded by a function of m. Let F(G) denote the Fitting subgroup of G and F,(G)/F(G) = F(G/F(G)). By Corollary 2.4, it sufficesto bound 1G : F(G)l. If e = G/F(G), thenF,(G) = C&@j) [4, 111.4.21and so it suffices to bound jF,(G)l. Hence we may assumethat G z=zF,(G). By Lemma 1.1(i), [ Cc(ct)l < m, and as c is now nilpotent, Corollary 2.4 gives us an a-invariant subgroup R ==:If/F(G) of (7, of nilpotency class at most two and index bounded by a function of m. Since F(G) = F(H), we may assumethat G -: H, so that G is nilpotent of classat most two. Let z :; Z/F(G) be the centre of G. Then e/Z is abelian and if G, = [G, LY]Z,then OLinverts every element of G,/Z, and Lemma 1.1(i) shows that I G : G, ) < m. Let 2 = A/F(G) be a maximal abelian normal subgroup of G, == GJF(G). Then G, 3 A 3 Z, and so A is a-invariant. Also 2 3 Ccl(A), and so bounding 12 I will bound : c1 1and hence c 1.Therefore we may assumethat (? is abelian, and since01inverts [c, 011, Lemma 1.1(i) allowsus to even assumethat (Yinverts %. Now let G, be the semidirect product G2 = G(+. Then F(G) ==F(G,) n G and so it suffices to show that 1G, : F(G&I is bounded. To this end, we can assumethat the Frattini subgroup of G2 is trivial [4, 111.4.21.Hence F(G,) is a direct product of minimal normal subgroupsof G, [4, 111.4.51,and so is F(G). Let W be such a minimal normal subgroup which is not centralized by [G, a]. We can view Was an irreducible KF-module, where k is a suitable finite prime field and F = G.JF(GJ, which is the split extension of an abelian normal subgroup B = GF(GJF(G,) g G/F(G) inverted by an involution which we may denote by a. If WI is any irreducible B-submodule of W, then B/C,( WI) is cyclic and C,(W,) 4 B(a). Since W = W, -i- Wla, we find that C,( WI) = C,(W), and so F/C,(W) IS . a dihedral group. We have [W, [B, a’JJ # 0 by hypothesis, and so W = [W, [B, a]] as W is irreducible. By Lemma 3.1, ( CW(~)!2= ( W i. Hence, if IV* is the product of all minimal normal subgroups like W, we have \ W* l2 = i C,+,.(a)!” ,< m2. Hence I[G, a] : C,,.,,(W*)( is bounded by a function of m. But CI,,,~( W*) = : Crc.~l(F(G)), clearly, and this is contained in F(G). Hence ([G, a]F(G)/F(G)I is bounded, and by a final application of Lemma I.l(i), we find that 1G/F(G)] is bounded.
REFERENCES BLACKBERN A?*‘D B. HIJPPERT, “Finite Groups II.” Springer-Verlag, Heidelberg/New York, to appear. 2. R. BRAUER AND K. FOWLER, On groups of even order, Ann. of Math. 62 (1955), 3. P. FONG, On orders of finite groups and centralizers of p-elements, Osahn 13 (1976), 483-489. 1. N.
Berlin/ 565-583. J. Math.
CENTRALIZERS
4. R. H~wrenr,
OF
“Endliche Gruppen I.” Springer-Verlag, Berlin;‘New AND B. A. F. WEHRFRITZ, “Locally Finite Groups,” Amsterdam, 1973. 6. V. P. %JKOV. On periodic groups with an almost regular involution, I1 (1972), 470-493 (Russian); Algebra and Logic I I (1972), 26&272 5. 0.
H.
KECEL
291
INVOLUTIOSS
York, 1967. North-Holland, Algebra i Z.ogika (English).
Printed
in Hr/~vm~
OF ALGEBRA
Periodic
64,
Groups
285-291
(1980)
in which the Centralizer Bounded Order
of an Involution
Has
B. HARTLEY Department
of Mathematics,
of Manrhcster,
University
Manchester,
En&and
AND ‘RI. Math.
Institut
MEIXNER
der Universitci’t,
Communicated Received
1.
6300
Giessetl,
Germarty
by B. Huppert May
4, I979
INTRODUCTION
The properties of involutions in groups, and in particular the centralizers of involutions, often exert a very strong influence over the structure of the group. One of the best known illustrations of this is the celebrated theorem of Brauer and Fowler [2], which asserts that there are only finitely many finite simple groups in which the centralizer of an involution has given order. This has been refined by Fong [3] as follows: 1 (Fong). There exists a function fi(c) such that ij G is a finite containing an involution i such that 1 C,(i)’ < c, then G : O,,,(S(G))/ .<
THEOREM
group fd4
Here S(G) denotes the soluble p-nilpotent subgroup of G. On the other hand, for infinite remarkable result:
radical groups,
of G and O,,,(G) Sunkov
the largest
normal
[6] has proved the following
TIIICOREV 2 (Sunkov). Let G be a periodic group containing an involution i with Jinite centralizer. Then G contains a soluble normal subgroup of jnite index, and so is locally finite.
In this paper we shall prove a theorem though both are used in its proof.
which
extends both of these results,
285 0021-8693!‘80/050285-07%02.00,!0 Copyright s. 1380 by Academic Press, Inc. All rights of reproduction in any form rescrvcd.
286
HARTLEY
Ah-D
YEIXNER
THEOREM. There exists a junction j(m) such that if G is a periodic group containing an involution i with 1C,(i)1 = m, then G contains a nilpotent subgroup of class at most two and index ,< j(m).
We make no attempt to describe such a function jexplicitly, except in certain special cases. By .%rnkov’s Theorem, G is locally finite, and a routine inverse limit argument, applied to the set of finite subgroups of G containing i (see [5, Chap. 1, Sect. K]) allows us to reduce the proof of the theorem to the case when G is finite. Thus this paper is really about finite groups. By Fong’s Theorem, G then contains a soluble normal subgroup of bounded index, and so we can assume G is soluble. We could even assume G 2-nilpotent, but this will not be necessary. In proving the theorem, it will usually be more convenient to think of a finite soluble group G admitting an involutory automorphism (z such that ! Co(o~)( = m. We have to show that G has a nilpotent subgroup of class at most two and bounded index. We do this in two stages. First we prove the result when G is nilpotcnt, and then we show that the Fitting subgroup F(G) of G has bounded index. The second part of the argument can be carried out when two is replaced by an arbitrary prime and will be dealt with in that generality in a later publication, but because of the simplicity of the argument for the prime two, it seems worth giving it separately. The following fact will be useful: LEMMA 1.1. Let G be a finite group admitting an automorphism (Y, and let N be a normal a-invariant subgroup of G. Then 0) (ii)
I Gd4I Ij(i(or)l,
G I Cd4 I G 1) = 1, then CGIN(a) = Cc(~)N/N.
Proof. (i) The proof of this can be found for example in [1], but we give it for convenience. If K/N = CcIN(ti), then by replacing G by K we can obviously assume that OLoperates trivially on G/N. Then the map 8: g --f [g, a] = g-igu maps G into N, and so 1im 8 1 < / N I. If g, , g, E G, then O(gi) = O(ga) if and only if gag,’ E C’,(z) =: C. Hence I im 0 1 = 1G : C j = : G I/i C I < 1N 1, and so j G : N ) = I G I/I N ! < I C /, as required. (ii)
This is well known,
see for example [4, 1.18.6J
Lemma 1.l in fact extends immediately to the locally finite case. 2.
NILPOTENT
GROUPS
LEMMA 2.1. Let G be afinitep-group ojclass 3 and suppose that 1 G’/y&G)/ p”. Then G contains a subgroup U such that
-
CENTRALIZERS
(i)
OF
287
INVOLUTIONS
C: > G’,
(ii)
1 G : U 1
(iii)
CT is nilpotent of class ai mosttwo.
We use {r,(G)}
to denote the lower central series of a group G, and G’ y--.y*(G).
Proof. Let Z(G) be the centre of G and Zi :- G’ n Z(G) 3 ya(G). Thus IG’:Z,I
inverts a group,
if it inverts
LEMMA 2.2. Let G beafinite p-group admitting an involutory automorphism LY. IfG L--:[G, a], then a:inverts ~,(G)/Y+~(G) if i isodd, and centralizes~(G)/Y~+~(G) if i is even.
Proof. We may assume G #- 1, in which we see [Xl ,..., hence [Xl ,..., The p-group.
case p + 2. Since G -=: [G, a], that 01inverts G/y,(G). If xi ,..., xi E G, then the coset of the commutator xi] modulo yiT1(G) depends only on the cosets of xi ,..., Xi mod G’ and mod yi+i(G) we have [xi ,..., xi]” 7-1 [xi” ,..., .qa] :+ [XL’ ,..., A-;‘] F xi]t-I)‘, from which the assertion follows. next lemma deals with the special case of the theorem when G is a When p + 2, notice that we obtain an explicit bound.
I,EMMA 2.3. Let G be a jnite p-group admitting an involutory automorphism cywith / Co(o)1= p”l. Then
(i) If p # 2, G contains an a-invariant subgroupof nilpotency classat mosttwo and index at mostpfW, wheref2(m) == la $ 22 -j- .** + m2. (ii) If p == 2, then G containsan o-invariant abelian subgroupof index at most2fJnr), wheref3(m) is a certain integer-valuedfunction.
Proof. (i) We use induction on m. If m = 0 then a! operates fixed point freely on G, and so G is abelian. Thus we may put f*(O) = 0. Now suppose that m > 0 and the statement has been proved for smaller values. If G > [G, a], then by Lemma 1.l(ii) we have I G : [G, a]1 := pi, where 0 < i < m, and for G, = [G, a] we have ] Ccl(~)1 = P”‘--~. By induction, G, contains an or-invariant subgroup of nilpotency class at most two and index at most pfkJm .
288
HARTLEY
AND
MEIXNER
Kow suppose that G [G, a]. Supp osc that G has nilpotency class t, and put s = t if t is odd, or s =I t J- 1 if t is even. Then, by Lemma 2.2, a inverts ys(G). We may suppose that s :l 1, and hence WC may choose an element .v E rs-i(G)\rz(G) such that xv E rJ(G). By Lemma 2.2, (Y centralizes subgroup of G. r,-l(G)lrs(G), and so H ---=
Finally, f.(m - 1) + m : j(j - 1) < fa(m - 1) + m + m(m f2(m - 1) 7 m* - fa(m), and so 1’ is the subgroup sought.
1) =
(ii) Here G is a 2-group, and by forming the semidirect product G(a), we can suppose that OLis an involution in G. Let A be a maximal abelian normal subgroup of G. If A, = Q,(A) = {a E A : a2 = I}, then thinking of A, as a vector space over the field of two elements and (Y as a linear transformation - 1) of A, > we have (a - I)* = (Y? - 2cr $- 1 = 0. Hence the map a+a(a. determines an embedding of AliCA1(a) into C,L((Y), and we have 1 A, 1 < Pm. Therefore A has rank at most 2~2, that is, .4 is a direct product of at most 2m cyclic groups. By Lemma 1.1(i), 1 A : [A, a]1 .< 2”‘, and so [A, a] 3 B = {Us”: a E A}. If w E A, then clearly cy inverts W-W --: [w, x]. Hence (Y inverts B. Since the inverting automorphism lies in the centre of the automorphism group of every abelian group, we find that [G, LY] centralizes B. By Lemma 1.1, C,(B), which contains [G, or], has index at most 2” in G. Since A/B has bounded rank and bounded exponent, its order is bounded in terms of m, and hence its centralizer in G has bounded index. Thus G contains a subgroup U of bounded index which centralizes both A/B and B. Hence U/C,(A) is isomorphic to a subgroup of Hom(A/B, B). This is because, if II E U, the map T,,: a + [u, u] is a homomorphism of A into B which induces one of A/B into B, and the map u--f 7” is the desired embedding. Now A/B is a direct product of at most 2m cyclic groups of orders dividing 2”‘, and B has rank at most 2m, and hence ) Hom(AIB, B)I = 1Hom(A/B, Q,,,(B))1 is at most 2.iff13.Hence I U : C,(A)1 is bounded by a function of m. Since A is a maximal abelian normal subgroup of G, C,(.4) 3.; 4, and hence I G : .4 is bounded in terms of m.
CENTRALIZERS
289
OF INVOLUTIONS
COROLLARY 2.4. Let G be a finite nilpotent group admitting an involutory automorphismLX,and let ( Co(a)1 : = m. Then G has an cu-invariant subgroupof nilpotency classat mosttwo and index boundedby a function of m.
Proof. We note that if p is any prime not dividing m, then (Y operates fixed point freely on the Sylow p-subgroup G, of G, and so G, is abelian. I,emma 2.3 then gives the result. 1Ve should remark for completeness that in the situation of the corollary, there need not be an abelian subgroup of bounded index. This is because an cy of extraspecial group P of order p2”i-l (p #: 2) admits an automorphism order two which inverts the Frattini factor group; thus 1 C,(a)1 =p, while k may be arbitrarily large.
3. GENERAL We need the following
simple
CASF
representation-theoretic
lemma:
LEMMA 3.1. Let G = A(u) be the semidirectproduct of a $nite abelian normal subgroupA by a cyclic group (u) of prime order q. Let k be a jeld and V a kc-module, ofjnite dimensionover k, suchthat I’ : [I’, [A, u]]. Thendim, V r q dim, C,,(u). Proof. If k is any extension field of k and Y = v Ok k, then C,(u) :.C,(u) @k k. This well known fact is proved by writing v = $&o V @ w, where a is a k-basis for k Hence we may assume that k is algebraically closed. Let p :? 0 be the characteristic of k, and write A = P x Q, where P is the Sylow p-subgroup of .4 and Q the Sylow p’-subgroup; put P =: 1 and Q 1 A if p 0. We claim that I’ = [V, [Q, u]]. For [I’, [Q, u]] is a KG-submodule of I’. If it is proper, then it is contained in a maximal submodule W of I/. The irreducible module I,‘/ W is trivial for P and for [Q, a] and hence for [A, u] = [P, u] x [Q, u]. Hence [V, [A, u]] < W, contrary to the hypothesis. Thus we indeed have I - = [I;, [Q, u]], and by considering Q(u> instead of G, we may assume that A is a p’-group. Then I’, is completely reducible, and W [II’, [,4, u]] for any kA-submodule W of I,-. Sow cvcry irreducible kG-module X such that X = [X, [A, u]] is induced from .4 and so has dimension q. For if X, is an irreducible A-submodule of X, then the stabilizer of X, is .-I, since otherwise X,, is homogenous, A acts on it by scalar multiplication and [.4, u] is trivial on X. Clifford’s Theorem then shows that X is induced from A, as claimed. It follows from this that if T is any irreducible k-4-submodule of r, so that dim, T = 1, then ?’ 3 Tu (+ ... 5; Tu’ ’ is an irreducible KG-submodule. Therefore I’ is a sum of irreducihlc KG-modules, each induced from A, and must be the direct sum of a collection of them. Each of thcsc contributes one to dim,: C,(U), and the result follows.
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Proof of theorem. By the remarks following the statement of the theorem, we need to consider a finite soluble group G admitting an involutory automorphism (Ysuch that ! C,(a)1 = m, and have to show that G contains a nilpotent subgroup of classat most two and index bounded by a function of m. Let F(G) denote the Fitting subgroup of G and F,(G)/F(G) = F(G/F(G)). By Corollary 2.4, it sufficesto bound 1G : F(G)l. If e = G/F(G), thenF,(G) = C&@j) [4, 111.4.21and so it suffices to bound jF,(G)l. Hence we may assumethat G z=zF,(G). By Lemma 1.1(i), [ Cc(ct)l < m, and as c is now nilpotent, Corollary 2.4 gives us an a-invariant subgroup R ==:If/F(G) of (7, of nilpotency class at most two and index bounded by a function of m. Since F(G) = F(H), we may assumethat G -: H, so that G is nilpotent of classat most two. Let z :; Z/F(G) be the centre of G. Then e/Z is abelian and if G, = [G, LY]Z,then OLinverts every element of G,/Z, and Lemma 1.1(i) shows that I G : G, ) < m. Let 2 = A/F(G) be a maximal abelian normal subgroup of G, == GJF(G). Then G, 3 A 3 Z, and so A is a-invariant. Also 2 3 Ccl(A), and so bounding 12 I will bound : c1 1and hence c 1.Therefore we may assumethat (? is abelian, and since01inverts [c, 011, Lemma 1.1(i) allowsus to even assumethat (Yinverts %. Now let G, be the semidirect product G2 = G(+. Then F(G) ==F(G,) n G and so it suffices to show that 1G, : F(G&I is bounded. To this end, we can assumethat the Frattini subgroup of G2 is trivial [4, 111.4.21.Hence F(G,) is a direct product of minimal normal subgroupsof G, [4, 111.4.51,and so is F(G). Let W be such a minimal normal subgroup which is not centralized by [G, a]. We can view Was an irreducible KF-module, where k is a suitable finite prime field and F = G.JF(GJ, which is the split extension of an abelian normal subgroup B = GF(GJF(G,) g G/F(G) inverted by an involution which we may denote by a. If WI is any irreducible B-submodule of W, then B/C,( WI) is cyclic and C,(W,) 4 B(a). Since W = W, -i- Wla, we find that C,( WI) = C,(W), and so F/C,(W) IS . a dihedral group. We have [W, [B, a’JJ # 0 by hypothesis, and so W = [W, [B, a]] as W is irreducible. By Lemma 3.1, ( CW(~)!2= ( W i. Hence, if IV* is the product of all minimal normal subgroups like W, we have \ W* l2 = i C,+,.(a)!” ,< m2. Hence I[G, a] : C,,.,,(W*)( is bounded by a function of m. But CI,,,~( W*) = : Crc.~l(F(G)), clearly, and this is contained in F(G). Hence ([G, a]F(G)/F(G)I is bounded, and by a final application of Lemma I.l(i), we find that 1G/F(G)] is bounded.
REFERENCES BLACKBERN A?*‘D B. HIJPPERT, “Finite Groups II.” Springer-Verlag, Heidelberg/New York, to appear. 2. R. BRAUER AND K. FOWLER, On groups of even order, Ann. of Math. 62 (1955), 3. P. FONG, On orders of finite groups and centralizers of p-elements, Osahn 13 (1976), 483-489. 1. N.
Berlin/ 565-583. J. Math.
CENTRALIZERS
4. R. H~wrenr,
OF
“Endliche Gruppen I.” Springer-Verlag, Berlin;‘New AND B. A. F. WEHRFRITZ, “Locally Finite Groups,” Amsterdam, 1973. 6. V. P. %JKOV. On periodic groups with an almost regular involution, I1 (1972), 470-493 (Russian); Algebra and Logic I I (1972), 26&272 5. 0.
H.
KECEL
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INVOLUTIOSS
York, 1967. North-Holland, Algebra i Z.ogika (English).
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