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The Maximal Number of Orbits of a Permutation Group with Bounded Movement
Journal of Algebra 214, 625᎐630 Ž1999. Article ID jabr.1998.7704, available online at http:rrwww.idealibrary.com on
The Maximal Number of Orbits of a Permutation Group with Bounded Movement* Jung R. Cho Department of Mathematics, The Pusan National Uni¨ ersity, Pusan, Korea, 609-735
Pan Soo Kim Department of Mathematics Education, The Pusan National Uni¨ ersity of Education, Pusan, Korea 607-736
and Cheryl E. Praeger Department of Mathematics, The Uni¨ ersity of Western Australia, Nedlands, Western Australia 6907, Australia Communicated by Alexander Lubotzky Received July 7, 1997
1. INTRODUCTION Let G be a permutation group on a set ⍀ such that G has no fixed points in ⍀. Then G is said to have bounded mo¨ ement if, for some positive integer m and for all g g G and ⌫ : ⍀, the cardinality < ⌫ g _ ⌫ < is at most m. When the maximum of < ⌫ g _ ⌫ < over all g g G and ⌫ : ⍀ is equal to m, we say G has bounded mo¨ ement equal to m. The third author has investigated groups with bounded movement in w6x, using a fundamental result of B. H. Neumann w3x which was transformed by P. M. Neumann w4, Lemma 2.3x into a theorem about the separation of subsets of points under a group action Žsee w5x for a survey.. For a group G of bounded movement equal to m, and having no orbits of length 1, it was * We are grateful to Avinoam Mann and Peter Neumann for their comments and assistance. 625 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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proved in w6x that both the number of G-orbits and the length of each G-orbit are finite, with upper bounds which are linear in m. In particular it was shown there that the number of G-orbits is at most 2 m y 1. We present here a classification of the groups for which the bound 2 m y 1 is attained Žsee w5, Question 3x.. We shall say that an orbit of a permutation group is nontri¨ ial if its length is greater than 1. The groups described below are examples of permutation groups of bounded movement equal to m having exactly 2 m y 1 orbits, all of them nontrivial. EXAMPLE. Let m s 2 ry1 G 1, and let G [ Z2r . Then G has 2 r y 1 s 2 m y 1 subgroups of index 2, say H1 , . . . , H2 my1. For i s 1, . . . , 2 m y 1, my1 let ⍀ i denote the set of two cosets of Hi in G, and set ⍀ [ D 2is1 ⍀ i. Then G acts faithfully on ⍀ by right multiplication with 2 m y 1 orbits ⍀ 1 , . . . , ⍀ 2 my1 , each of length 2. Each nontrivial element g g G lies in exactly 2 ry1 y 1 s m y 1 of the subgroups Hi and permutes nontrivially the remaining m s 2 ry1 of the ⍀ i . Thus each nontrivial element of G has m s 2 ry1 cycles of length 2 in ⍀. For any subset ⌫ : ⍀ and any g g G, the set ⌫ g _ ⌫ consists of at most 1 point from each of the G-orbits on which g acts nontrivially, and hence < ⌫ g _ ⌫ < F m. It follows that G has bounded movement equal to m and G has 2 m y 1 nontrivial orbits in ⍀. It turns out that for m ) 1, these are the only examples meeting the bound. THEOREM 1. Let m be a positi¨ e integer, and suppose that G is a permutation group on a set ⍀ such that G has no fixed points in ⍀, G has bounded mo¨ ement equal to m, and G has 2 m y 1 orbits in ⍀. Then m is a power of 2, and either Ži.
m s 1, < ⍀ < s 3, and G is Z3 or S3 , or
Žii. G is one of the groups in the example abo¨ e, that is, G is elementary abelian of order 2 m, all G-orbits ha¨ e length 2, and the pointwise stabilizers of the G-orbits are precisely the 2 m y 1 distinct subgroups of G of index 2. It was pointed out to us by Avinoam Mann that Theorem 1 yields, as an immediate corollary, a classification of groups satisfying another extremal condition. In w6x it was shown that, for a permutation group G on a set ⍀, with no fixed points in ⍀, such that G has bounded movement equal to m, the size of ⍀ is at most 5m y 2. If < ⍀ < s 5m y 2 then it follows easily from the analysis in w6x that G has 2 m y 1 orbits in ⍀, and then Theorem 1 implies that m s 1 and G is Z3 or S3 . Thus we have the following corollary to Theorem 1.
GROUPS WITH BOUNDED MOVEMENT
627
COROLLARY TO THEOREM 1. Let m be a positi¨ e integer and suppose that G is a permutation group on a set ⍀, with no fixed points in ⍀, such that G has bounded mo¨ ement equal to m. Then < ⍀ < F 5m y 2 with equality if and only if m s 1 and G s Z3 or S3 . The classification in Theorem 1 in the case where m s 1 can be deduced immediately from the classification by Leonid Brailovsky w1x of subsets of movement 1 relative to some group action. For a positive integer m, a subset ⌫ : ⍀ is said to have mo¨ ement m relative to a permutation group G F SymŽ ⍀ . if < ⌫ g _ ⌫ < is bounded above, for g g G, and its maximum value is m. We write move Ž ⌫ . s m. Brailovsky w1x showed that a subset of ⍀ has movement 1 relative to G F SymŽ ⍀ . if and only if it is equal to a union of G-orbits in ⍀ with either one point added or one point removed. When m ) 1 the classification in Theorem 1 follows immediately from the following theorem about subsets with movement m. This theorem was proved and shown to us by Peter Neumann after seeing a draft of our paper which had been submitted for publication. On the suggestion of the editors and with Peter’s agreement we rewrote our paper incorporating his result. Our exposition uses the language of permutation groups, whereas Peter’s proof was written in terms of covers for abstract groups. Let G be a permutation group on a set ⍀ with orbits ⍀ i , for i g I. We shall say that a subset ⌫ : ⍀ cuts across each G-orbit if ⌫i [ ⌫ l ⍀ i f ⭋, ⍀ i 4 for all i g I. THEOREM 2. Let G F SymŽ ⍀ . be a permutation group with t orbits. Suppose that ⌫ : ⍀ is a set such that mo¨ eŽ ⌫ . s m ) 1, and such that ⌫ cuts across each G-orbit. Then t F 2 m y 1, and moreo¨ er if t s 2 m y 1 then Ž1. G is an elementary abelian 2-group and e¨ ery G-orbit has size 2; Ž2. if the rank of G is r then r G 2, t s 2 r y 1, and m s 2 ry1 ; Ž3. the t different G-orbits are Ž isomorphic to . the coset spaces of the 2 r y 1 different subgroups of index 2 in G. The assertion that t F 2 m y 1 was proved in w2, Lemma 3.5x. It is the characterization of the groups for which the upper bound is attained which concerns us here. From our observations above, all of our results will follow from this characterization. Thus the rest of the paper is devoted to proving it.
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2. PROOF OF THEOREM 2 Let m be a positive integer greater than 1. Suppose that G F SymŽ ⍀ . with orbits ⍀ 1 , ⍀ 2 , . . . , ⍀ t , where t s 2 m y 1. Suppose further that ⌫ : ⍀ has moveŽ ⌫ . s m and that ⌫ cuts across each of the G-orbits ⍀ i . For each i set n i [ < ⍀ i < and ⌫i [ ⌫ l ⍀ i . Note that 0 - < ⌫i < - n i . Claim. If Theorem 2 holds for the special case in which < ⌫i < s 1 for each i s 1, . . . , 2 m y 1, then it holds in general. Suppose that Theorem 2 holds for the case where each < ⌫i < s 1. For i s 1, . . . , t, define ⌺ i [ ⌫i g < g g G4 , and note that < ⌺ i < G 2 since ⌫ cuts across ⍀ i . Set ⌺ [ D i G 1 ⌺ i . Then G induces a natural action on ⌺ for which the G-orbits are ⌺ 1 , . . . , ⌺ t . Let G ⌺ denote the permutation group induced by G on ⌺, and let K denote the kernel of this action. We claim that the t-element subset ⌫⌺ s ⌫1 , . . . ⌫t 4 : ⌺ has movement equal to m relative to G ⌺ , and that ⌫⌺ cuts across each G ⌺-orbit ⌺ i . For each g g G, < ⌫ g _ ⌫ < F m and hence < ⌫⌺g _ ⌫⌺ < F m. Thus moveŽ ⌫⌺ . F m. Also, since < ⌺ i < G 2 and ⌫⌺ l ⌺ i consists of the single element ⌫i of ⌺ i , the set ⌫⌺ cuts across each of the 2 m y 1 orbits ⌺ i . However, it follows from w2, Lemma 3.5x that the number of G ⌺ -orbits is at most 2 ⭈ moveŽ ⌫⌺ . y 1, and hence moveŽ ⌫⌺ . s m. Thus the hypotheses of Theorem 2 hold for the subset ⌫⌺ : ⌺ relative to G ⌺ , and ⌫⌺ meets each G ⌺-orbit in exactly one point. By our assumption it follows that t s 2 r y 1 s 2 m y 1 for some r ) 1, and that G ⌺ s Z2r and each < ⌺ i < s 2. Further, the subgroups Hi of G fixing ⌫i setwise range over the 2 r y 1 distinct subgroups which have index 2 in G and which contain K. In particular, for each i, Hi is normal in G and hence the Hi-orbits in ⍀ i are blocks of imprimitivity for G, and their number is at most < G : Hi < s 2. Since Hi fixes ⌫i setwise it follows that ⌫i is an Hi-orbit and n i s 2 < ⌫i <. Let g g G _ K. Then in its action on ⌺, g moves exactly m of the ⌫i . Since the ⌫i are blocks of imprimitivity for G, each ⌫i g is equal to either ⌫i or ⍀ i _ ⌫i . It follows that < ⌫ g _ ⌫ < is equal to the sum of the sizes of the m subsets ⌫i moved by g. However, since moveŽ ⌫ . s m, each of these m subsets ⌫i must have size 1. Since for each i we may choose an element g which moves ⌫i , we deduce that each of the ⌫i has size 1, and that K is the identity subgroup. It follows that Theorem 2 holds for G. Thus the claim is proved. From now on we may and shall assume that each < ⌫i < s 1. Let ⌫i s i 4 . Further we may assume that n1 F n 2 F ⭈⭈⭈ F n t . For g g G let cŽ g . denote the number of integers i such that ig s i . Note that since moveŽ ⌫ . s m we have cŽ g . G t y m s m y 1, and also cŽ1 G . s t ) m y 1. Next we show that at least one of the ⍀ i has length 2.
629
GROUPS WITH BOUNDED MOVEMENT
LEMMA 2.1.
n1 s 2.
Proof. Let X denote the number of pairs Ž g, i . such that g g G, 1 F i F t, and ig s i . Then X s Ý g g G cŽ g ., and by our observations, X ) < G < ⭈ Ž m y 1.. On the other hand, for each i, the number of elements of G which fix i is < G i < s < G
Ý
c Ž g . G < G _ H < ⭈ Ž m y 1. s < G < ⭈
ggG_H
my1 2
.
On the other hand, for each i G 2, the number of elements of G _ H which fix i is < G i _ H <. If G i s H then < G i _ H < s 0, while if G i / H then < G i _ H < s < G i
Ys
Ý < G
_H
is2
2
t
Ý is2
1 ni
F
⭈ Ž t y 1. s < G < ⭈
my1 2
.
It follows that equality holds in both of the displayed approximations for Y. This means in particular that each n i s 2, whence G s Z2r for some r. Further, for each i G 2, G i / H and so r G 2. Arguing in the same way with H replaced by G i , for some i G 2, we see that G i / G j if j / i, and also if g f G i then cŽ g . s m y 1. Thus the stabilizers G i Ž1 F i F t . are pairwise distinct, and if g / 1 then cŽ g . s m y 1. Finally we determine m. LEMMA 2.3.
m s 2 ry1.
Proof. We use the information in Lemma 2.2 to determine precisely the quantity X s Ý g g G cŽ g .: X s t q Ž < G < y 1. ⭈ Ž m y 1. s 2 m y 1 q Ž 2 r y 1. Ž m y 1. .
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CHO, KIM, AND PRAEGER
On the other hand, from the proof of Lemma 2.1, t
X s < G < Ý ny1 s i is1
s 2 ry1 ⭈ Ž 2 m y 1 . .
Thus Ž2 ry1 y 1.Ž2 m y 1. s Ž2 r y 1.Ž m y 1.. Since the integers 2 r y 1 and 2 ry1 y 1 are relatively prime, 2 r y 1 divides 2 m y 1, and since the integers 2 m y 1 and m y 1 are relatively prime, 2 m y 1 divides 2 r y 1. It follows that 2 m y 1 s 2 r y 1, whence m s 2 ry1. The proof of Theorem 2 now follows from Lemmas 2.1᎐2.3.
REFERENCES 1. L. Brailovsky, Structure of quasi-invariant sets, Arch. Math. 59 Ž1992., 322᎐326. 2. L. Brailovsky, D. Pasechnik, and C. E. Praeger, Subsets close to invariant subsets for group actions, Proc. Amer. Math. Soc. 123 Ž1995., 2283᎐2295. 3. B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 Ž1954., 236᎐248. 4. P. M. Neumann, The structure of finitary permutation groups, Arch. Math. Ž Basel . 27 Ž1976., 3᎐17. 5. C. E. Praeger, The separation theorem for group actions, in ‘‘Ordered Groups and Infinite Groups’’ ŽW. Charles Holland, Ed.., Kluwer Academic, DordrechtrBostonrLondon, 1995. 6. C. E. Praeger, On permutation groups with bounded movement, J. Algebra 144 Ž1991., 436᎐442.
The Maximal Number of Orbits of a Permutation Group with Bounded Movement* Jung R. Cho Department of Mathematics, The Pusan National Uni¨ ersity, Pusan, Korea, 609-735
Pan Soo Kim Department of Mathematics Education, The Pusan National Uni¨ ersity of Education, Pusan, Korea 607-736
and Cheryl E. Praeger Department of Mathematics, The Uni¨ ersity of Western Australia, Nedlands, Western Australia 6907, Australia Communicated by Alexander Lubotzky Received July 7, 1997
1. INTRODUCTION Let G be a permutation group on a set ⍀ such that G has no fixed points in ⍀. Then G is said to have bounded mo¨ ement if, for some positive integer m and for all g g G and ⌫ : ⍀, the cardinality < ⌫ g _ ⌫ < is at most m. When the maximum of < ⌫ g _ ⌫ < over all g g G and ⌫ : ⍀ is equal to m, we say G has bounded mo¨ ement equal to m. The third author has investigated groups with bounded movement in w6x, using a fundamental result of B. H. Neumann w3x which was transformed by P. M. Neumann w4, Lemma 2.3x into a theorem about the separation of subsets of points under a group action Žsee w5x for a survey.. For a group G of bounded movement equal to m, and having no orbits of length 1, it was * We are grateful to Avinoam Mann and Peter Neumann for their comments and assistance. 625 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
626
CHO, KIM, AND PRAEGER
proved in w6x that both the number of G-orbits and the length of each G-orbit are finite, with upper bounds which are linear in m. In particular it was shown there that the number of G-orbits is at most 2 m y 1. We present here a classification of the groups for which the bound 2 m y 1 is attained Žsee w5, Question 3x.. We shall say that an orbit of a permutation group is nontri¨ ial if its length is greater than 1. The groups described below are examples of permutation groups of bounded movement equal to m having exactly 2 m y 1 orbits, all of them nontrivial. EXAMPLE. Let m s 2 ry1 G 1, and let G [ Z2r . Then G has 2 r y 1 s 2 m y 1 subgroups of index 2, say H1 , . . . , H2 my1. For i s 1, . . . , 2 m y 1, my1 let ⍀ i denote the set of two cosets of Hi in G, and set ⍀ [ D 2is1 ⍀ i. Then G acts faithfully on ⍀ by right multiplication with 2 m y 1 orbits ⍀ 1 , . . . , ⍀ 2 my1 , each of length 2. Each nontrivial element g g G lies in exactly 2 ry1 y 1 s m y 1 of the subgroups Hi and permutes nontrivially the remaining m s 2 ry1 of the ⍀ i . Thus each nontrivial element of G has m s 2 ry1 cycles of length 2 in ⍀. For any subset ⌫ : ⍀ and any g g G, the set ⌫ g _ ⌫ consists of at most 1 point from each of the G-orbits on which g acts nontrivially, and hence < ⌫ g _ ⌫ < F m. It follows that G has bounded movement equal to m and G has 2 m y 1 nontrivial orbits in ⍀. It turns out that for m ) 1, these are the only examples meeting the bound. THEOREM 1. Let m be a positi¨ e integer, and suppose that G is a permutation group on a set ⍀ such that G has no fixed points in ⍀, G has bounded mo¨ ement equal to m, and G has 2 m y 1 orbits in ⍀. Then m is a power of 2, and either Ži.
m s 1, < ⍀ < s 3, and G is Z3 or S3 , or
Žii. G is one of the groups in the example abo¨ e, that is, G is elementary abelian of order 2 m, all G-orbits ha¨ e length 2, and the pointwise stabilizers of the G-orbits are precisely the 2 m y 1 distinct subgroups of G of index 2. It was pointed out to us by Avinoam Mann that Theorem 1 yields, as an immediate corollary, a classification of groups satisfying another extremal condition. In w6x it was shown that, for a permutation group G on a set ⍀, with no fixed points in ⍀, such that G has bounded movement equal to m, the size of ⍀ is at most 5m y 2. If < ⍀ < s 5m y 2 then it follows easily from the analysis in w6x that G has 2 m y 1 orbits in ⍀, and then Theorem 1 implies that m s 1 and G is Z3 or S3 . Thus we have the following corollary to Theorem 1.
GROUPS WITH BOUNDED MOVEMENT
627
COROLLARY TO THEOREM 1. Let m be a positi¨ e integer and suppose that G is a permutation group on a set ⍀, with no fixed points in ⍀, such that G has bounded mo¨ ement equal to m. Then < ⍀ < F 5m y 2 with equality if and only if m s 1 and G s Z3 or S3 . The classification in Theorem 1 in the case where m s 1 can be deduced immediately from the classification by Leonid Brailovsky w1x of subsets of movement 1 relative to some group action. For a positive integer m, a subset ⌫ : ⍀ is said to have mo¨ ement m relative to a permutation group G F SymŽ ⍀ . if < ⌫ g _ ⌫ < is bounded above, for g g G, and its maximum value is m. We write move Ž ⌫ . s m. Brailovsky w1x showed that a subset of ⍀ has movement 1 relative to G F SymŽ ⍀ . if and only if it is equal to a union of G-orbits in ⍀ with either one point added or one point removed. When m ) 1 the classification in Theorem 1 follows immediately from the following theorem about subsets with movement m. This theorem was proved and shown to us by Peter Neumann after seeing a draft of our paper which had been submitted for publication. On the suggestion of the editors and with Peter’s agreement we rewrote our paper incorporating his result. Our exposition uses the language of permutation groups, whereas Peter’s proof was written in terms of covers for abstract groups. Let G be a permutation group on a set ⍀ with orbits ⍀ i , for i g I. We shall say that a subset ⌫ : ⍀ cuts across each G-orbit if ⌫i [ ⌫ l ⍀ i f ⭋, ⍀ i 4 for all i g I. THEOREM 2. Let G F SymŽ ⍀ . be a permutation group with t orbits. Suppose that ⌫ : ⍀ is a set such that mo¨ eŽ ⌫ . s m ) 1, and such that ⌫ cuts across each G-orbit. Then t F 2 m y 1, and moreo¨ er if t s 2 m y 1 then Ž1. G is an elementary abelian 2-group and e¨ ery G-orbit has size 2; Ž2. if the rank of G is r then r G 2, t s 2 r y 1, and m s 2 ry1 ; Ž3. the t different G-orbits are Ž isomorphic to . the coset spaces of the 2 r y 1 different subgroups of index 2 in G. The assertion that t F 2 m y 1 was proved in w2, Lemma 3.5x. It is the characterization of the groups for which the upper bound is attained which concerns us here. From our observations above, all of our results will follow from this characterization. Thus the rest of the paper is devoted to proving it.
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CHO, KIM, AND PRAEGER
2. PROOF OF THEOREM 2 Let m be a positive integer greater than 1. Suppose that G F SymŽ ⍀ . with orbits ⍀ 1 , ⍀ 2 , . . . , ⍀ t , where t s 2 m y 1. Suppose further that ⌫ : ⍀ has moveŽ ⌫ . s m and that ⌫ cuts across each of the G-orbits ⍀ i . For each i set n i [ < ⍀ i < and ⌫i [ ⌫ l ⍀ i . Note that 0 - < ⌫i < - n i . Claim. If Theorem 2 holds for the special case in which < ⌫i < s 1 for each i s 1, . . . , 2 m y 1, then it holds in general. Suppose that Theorem 2 holds for the case where each < ⌫i < s 1. For i s 1, . . . , t, define ⌺ i [ ⌫i g < g g G4 , and note that < ⌺ i < G 2 since ⌫ cuts across ⍀ i . Set ⌺ [ D i G 1 ⌺ i . Then G induces a natural action on ⌺ for which the G-orbits are ⌺ 1 , . . . , ⌺ t . Let G ⌺ denote the permutation group induced by G on ⌺, and let K denote the kernel of this action. We claim that the t-element subset ⌫⌺ s ⌫1 , . . . ⌫t 4 : ⌺ has movement equal to m relative to G ⌺ , and that ⌫⌺ cuts across each G ⌺-orbit ⌺ i . For each g g G, < ⌫ g _ ⌫ < F m and hence < ⌫⌺g _ ⌫⌺ < F m. Thus moveŽ ⌫⌺ . F m. Also, since < ⌺ i < G 2 and ⌫⌺ l ⌺ i consists of the single element ⌫i of ⌺ i , the set ⌫⌺ cuts across each of the 2 m y 1 orbits ⌺ i . However, it follows from w2, Lemma 3.5x that the number of G ⌺ -orbits is at most 2 ⭈ moveŽ ⌫⌺ . y 1, and hence moveŽ ⌫⌺ . s m. Thus the hypotheses of Theorem 2 hold for the subset ⌫⌺ : ⌺ relative to G ⌺ , and ⌫⌺ meets each G ⌺-orbit in exactly one point. By our assumption it follows that t s 2 r y 1 s 2 m y 1 for some r ) 1, and that G ⌺ s Z2r and each < ⌺ i < s 2. Further, the subgroups Hi of G fixing ⌫i setwise range over the 2 r y 1 distinct subgroups which have index 2 in G and which contain K. In particular, for each i, Hi is normal in G and hence the Hi-orbits in ⍀ i are blocks of imprimitivity for G, and their number is at most < G : Hi < s 2. Since Hi fixes ⌫i setwise it follows that ⌫i is an Hi-orbit and n i s 2 < ⌫i <. Let g g G _ K. Then in its action on ⌺, g moves exactly m of the ⌫i . Since the ⌫i are blocks of imprimitivity for G, each ⌫i g is equal to either ⌫i or ⍀ i _ ⌫i . It follows that < ⌫ g _ ⌫ < is equal to the sum of the sizes of the m subsets ⌫i moved by g. However, since moveŽ ⌫ . s m, each of these m subsets ⌫i must have size 1. Since for each i we may choose an element g which moves ⌫i , we deduce that each of the ⌫i has size 1, and that K is the identity subgroup. It follows that Theorem 2 holds for G. Thus the claim is proved. From now on we may and shall assume that each < ⌫i < s 1. Let ⌫i s i 4 . Further we may assume that n1 F n 2 F ⭈⭈⭈ F n t . For g g G let cŽ g . denote the number of integers i such that ig s i . Note that since moveŽ ⌫ . s m we have cŽ g . G t y m s m y 1, and also cŽ1 G . s t ) m y 1. Next we show that at least one of the ⍀ i has length 2.
629
GROUPS WITH BOUNDED MOVEMENT
LEMMA 2.1.
n1 s 2.
Proof. Let X denote the number of pairs Ž g, i . such that g g G, 1 F i F t, and ig s i . Then X s Ý g g G cŽ g ., and by our observations, X ) < G < ⭈ Ž m y 1.. On the other hand, for each i, the number of elements of G which fix i is < G i < s < G
Ý
c Ž g . G < G _ H < ⭈ Ž m y 1. s < G < ⭈
ggG_H
my1 2
.
On the other hand, for each i G 2, the number of elements of G _ H which fix i is < G i _ H <. If G i s H then < G i _ H < s 0, while if G i / H then < G i _ H < s < G i
Ys
Ý < G
_H
is2
2
t
Ý is2
1 ni
F
⭈ Ž t y 1. s < G < ⭈
my1 2
.
It follows that equality holds in both of the displayed approximations for Y. This means in particular that each n i s 2, whence G s Z2r for some r. Further, for each i G 2, G i / H and so r G 2. Arguing in the same way with H replaced by G i , for some i G 2, we see that G i / G j if j / i, and also if g f G i then cŽ g . s m y 1. Thus the stabilizers G i Ž1 F i F t . are pairwise distinct, and if g / 1 then cŽ g . s m y 1. Finally we determine m. LEMMA 2.3.
m s 2 ry1.
Proof. We use the information in Lemma 2.2 to determine precisely the quantity X s Ý g g G cŽ g .: X s t q Ž < G < y 1. ⭈ Ž m y 1. s 2 m y 1 q Ž 2 r y 1. Ž m y 1. .
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On the other hand, from the proof of Lemma 2.1, t
X s < G < Ý ny1 s i is1
s 2 ry1 ⭈ Ž 2 m y 1 . .
Thus Ž2 ry1 y 1.Ž2 m y 1. s Ž2 r y 1.Ž m y 1.. Since the integers 2 r y 1 and 2 ry1 y 1 are relatively prime, 2 r y 1 divides 2 m y 1, and since the integers 2 m y 1 and m y 1 are relatively prime, 2 m y 1 divides 2 r y 1. It follows that 2 m y 1 s 2 r y 1, whence m s 2 ry1. The proof of Theorem 2 now follows from Lemmas 2.1᎐2.3.
REFERENCES 1. L. Brailovsky, Structure of quasi-invariant sets, Arch. Math. 59 Ž1992., 322᎐326. 2. L. Brailovsky, D. Pasechnik, and C. E. Praeger, Subsets close to invariant subsets for group actions, Proc. Amer. Math. Soc. 123 Ž1995., 2283᎐2295. 3. B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc. 29 Ž1954., 236᎐248. 4. P. M. Neumann, The structure of finitary permutation groups, Arch. Math. Ž Basel . 27 Ž1976., 3᎐17. 5. C. E. Praeger, The separation theorem for group actions, in ‘‘Ordered Groups and Infinite Groups’’ ŽW. Charles Holland, Ed.., Kluwer Academic, DordrechtrBostonrLondon, 1995. 6. C. E. Praeger, On permutation groups with bounded movement, J. Algebra 144 Ž1991., 436᎐442.