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Sylow Subgroups Which Are Maximal in the Universal Locally Finite Group of Philip Hall

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Sylow Subgroups Which Are Maximal in the Universal Locally Finite Group of Philip Hall Journal of Algebra 215, 229᎐234Ž. 1999 Article ID jabr.1998.7685, available online at http:rrwww.idealibrary.com on Sylow Subgroups Which Are Maximal in the Universal Locally Finite Group of Philip Hall Marianna Dalle Molle View metadata, citation and similar papers at core.ac.uk brought to you by CORE ¨ ¨ ¨ Dipartimento di Matematica Pura ed Applicata, Uni ersita di Pado a, ia Belzoni 7, provided by Elsevier - Publisher Connector 35131 Padua, Italy E-mail: [email protected] Communicated by Peter M. Neumann Received November 18, 1997 For every prime p, we construct a subgroup of Philip Hall's universal locally finite group which is both maximal and a p-group. This provides an example of a simple locally finite group with a maximal subgroup which is locally nilpotent. ᮊ 1999 Academic Press 1. INTRODUCTION In this paper we prove that for each prime p, P. Hall's universal countable locally finite groupŽ seewx 4. has a maximal subgroup that is a p-group, thus providing an example of an infinite locally finite simple group that contains maximal, locally nilpotent p-subgroups. Therefore, Thompson's theorem stating that a finite group which contains a maximal nilpotent subgroup of odd order is solvableŽ seewx 5, Theorem 10.4.2 . cannot be extended to prove the local solvability of locally finite groups containing maximal, locally nilpotent subgroups. One generalization of Thompson's result to linear groups was given by Wehrfritz inw 6, Theorem 12.8x : a periodic linear group G with a maximal locally nilpotent subgroup M is solvable provided the Sylow subgroups of M fulfill certain conditions. wx r In 1 these conditions were relaxed to require that either M coreGŽ.M is not a 2-group or it is nilpotent of class at most 2. The simple group PSLŽ. 2, 17 , which has a dihedral group of order 16 and nilpotency class 3 as maximal subgroup, shows that the restrictions are necessary in the case r wx that M coreGŽ.M is a 2-group. Bruno and Schuur in 2 extended this theme to a locally finite group G, finding among other things that G is 229 0021-8693r99 $30.00 Copyright ᮊ 1999 by Academic Press All rights of reproduction in any form reserved. 230 MARIANNA DALLE MOLLE solvable if it contains a maximal subgroup M satisfying one of the following conditions: r Ž.i M is nilpotent and M coreGŽ.M is not a p-group for any prime p, Ž.ii M is nilpotent of class at most 2, Ž.iii M is nilpotent and has min-p for the odd primes p involved in M, and the Sylow 2-subgroup of M has nilpotency class at most 2. The main result of this paper is the following: THEOREM 1.1. The countable uni¨ersal locally finite group U constructed by Philip HallŽ see wx4 . contains, for each prime p, a maximal subgroup that is ap-group. We recall the definition of the universal group U. Let G1 be the symmetric group on 3 elements. Define a sequence of groups Gn and ␦ ª embeddings nn: G G nq1 inductively in the following way: Gnq1 is the ␦ symmetric group on the set Gnnand is the right regular representation of G into SymŽ.G . We define U to be the direct limit lim Ä4G , ␦ . nn ª nn ␦ Generally, we will identify Gnnwith the subgroup Ž.Gnof Gnq1, so that U can be thought of as the union of all groups Gn. 2. REDUCTION For the remainder of the paper, let p be a fixed prime. Note 2.1. Clearly the definition of U is not affected if we omit finitely many terms of the series Ä4Gn . So, for technical reasons, we may assume s that the first term of the series Ä4Gn defined above is G1 SymŽ.k , where ) X k 5p. Consider now the series ÄGn4 of derived subgroupsŽ recall that the X s D X Gnnare alternating groups. Then U Gn, since U is simple. Moreover, ␦ X g Gnnis embedded via in the alternating group Gnq1, since for all x Gn < < ␦ the index Gnn: ²:x is even and so Ž.x is the product of an even number of cycles of length <²:x <. Starting from the chain of groups Gn described in Note 2.1, we will find a chain Ä4Pnn, where P is a Sylow p-subgroup of Gn, such that the D p-subgroup nnP is a maximal subgroup of U. The key for the construc- tion of the groups Pn lies in the following proposition whose proof is given in Sections 3 and 4. ¨ PROPOSITION 2.2. For any natural number n, gi en a Sylow p-subgroup Pn g _ of Gnn and x G Pn, there exists a Sylow p-subgroup Pnq1 of Gnq1 such - G X that PnnPq1 and²: x, Pnq1 Gnq1. SYLOW SUBGROUPS 231 Obser¨ation 2.3. Proposition 2.2 implies Theorem 1.1. g ގ Proof. Let k and Pkkbe a Sylow p-subgroup of G . Then there is a s G q - - иии - natural number m fkŽ. k 1 and a sequence PkkP q1 - F F Pmy1 Pmj, where P is a Sylow p-subgroup of Gjfor k j m, such that G X g _ _ s ²:x, PmG mfor all x GkkP : for, let GkkP Ä4x12, x ,..., x s . Us- ing Proposition 2.2, choose Pkq1, a Sylow p-subgroup of Gkq1, such that - G X g _ PkkP q11and ²:x , Pkq1 Gkq12. Since x Gkq1 Pkq1, again by g - Proposition 2.2, we can choose Pkq2 Syl pkŽ.G q2 such that Pkq1 Pkq2 G X and ²:x2 , Pkq2 Gkq2 . Repeat the construction and get that Pkqi is a - G X Sylow p-subgroup of Gkqiksuch that P qiy1 Pkqiikand ²:x , P qikG qi for all i s 1,...,s. Set m s fkŽ.s k q s. For i s 1,...,s, we have X X ²:²x , P G x , P q , P :²G G q , P :²G x , P :G G , since x is in im ikimX kim sm mX s Gkkwhich is embedded in G q1 Ž.see Note 2.1 and so also in Gkqi. Now, fix s s a Sylow p-subgroup P11of G , set m 01 and mttfmŽ.y1 , and consider - the sequence of Sylow p-subgroups Pjjof G as constructed above: P12P - иии - P - P q - иии - P - иии - P - иии . By construction mm11X 1 mm2 t ²:x, P G G for all t and x g G _ P . Set M s D P . We claim mmtt mmty1 ty1 ii that M is a maximal subgroup of U.Sofor x g U _ M and g g U we have to prove that g g ²:x, M . Since x g U and x f M, there exists a natural number l such that x g G _ P for all h G l. By Note 2.1, since g g U, hh X there is a natural number k such that m y G l and g g G . Then X k 1 m k x g G _ P and so ²:x, P G G . It follows that g g ²:x, M . mmky 1 ky1 mkk m A theorem of JordanŽ seewx 3, Theorem 3.3E. states that a primitive permutation group of degree n ) p q 2 containing a p-cycle is either alternating or symmetricŽ. the condition p a prime is necessary . Thus, X since any Sylow p-subgroup of Gnq1 contains a p-cycle and Gnq1 is an alternating group, in order to prove Proposition 2.2 it is enough to find for g _ each x GnnP a Sylow p-subgroup Pnq1 of Gnq1 such that ²:x, Pnq1 is transitive and primitive on Gn. The remainder of the paper is devoted to the proof of the existence of a Sylow p-subgroup Pnq1 of Gnq1 with the desired properties. 3. CRITERION Let G be a group acting on a set ⍀. Recall that ⌳ : ⍀ is a Jordan set for G on ⍀ if <<⌳ ) 1 and there exists a subgroup of G acting transitively on ⌳ and trivially on ⍀ _ ⌳. LEMMA 3.1wx 3, Lemma 7.4B . Suppose that a group G acts transiti¨ely and imprimiti¨ely on a set ⍀. Let ⌳ be a Jordan subset on ⍀ and let B be a nontri¨ial block for G such that B l ⌳ / л. Then either B : ⌳ or ⌳ : B. 232 MARIANNA DALLE MOLLE Proof. Suppose B ­ ⌳, and let b be an element in B _ ⌳ and ␭ g B l ⌳. Since ⌳ is a Jordan set, there is a subgroup H of G that fixes all elements of ⍀ _ ⌳ and is transitive on ⌳. Hence, for each h g H, we have bh s b g B l Bh, and therefore B s Bh. The transitivity of H on ⌳ implies ⌳ s ␭H, and so ⌳ ; B. The Sylow p-subgroups of the symmetric groups will play a fundamental part in the following proofs. So we recall their structure and properties. For more details seewx 3, Example 2.6.1 . Note 3.2. We first describe the structure of the Sylow p-subgroups of a symmetric group SymŽ.⍀ over a finite set ⍀ of order p s with s G 1. Under this assumption a Sylow p-subgroup Q of SymŽ.⍀ is isomorphic to иии the iterated wreath product Cppwr C wr wr C pwith s factors. Since the wreath product of two transitive permutation groups is transitive on the cartesian product of their supports, the group Q is transitive on ⍀ by <<⍀ s a ¦ s ␣ induction.
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