Journal of Algebra 228, 367᎐376Ž. 2000 doi:10.1006rjabr.2000.8269, available online at http:rrwww.idealibrary.com on

A Note on Frobenius Groups

Paul Flavell

The School of Mathematics and Statistics, The Uni¨ersity of Birmingham, CORE Birmingham B15 2TT, United Kingdom Metadata, citation and similar papers at core.ac.uk E-mail: [email protected] Provided by Elsevier - Publisher Connector Communicated by George Glauberman

Received January 25, 1999

1. INTRODUCTION

Two celebrated applications of the character theory of finite groups are Burnside’s p␣q ␤-Theorem and the theorem of Frobenius on the groups that bear his name. The elegant proofs of these theorems were obtained at the beginning of this century. It was then a challenge to find character-free proofs of these results. This was achieved for the p␣q ␤-Theorem by Benderwx 1 , Goldschmidt wx 3 , and Matsuyama wx 8 in the early 1970s. Their proofs used ideas developed by Feit and Thompson in their proof of the Odd Order Theorem. There is no known character-free proof of Frobe- nius’ Theorem. Recently, Corradi´´ and Horvathwx 2 have obtained some partial results in this direction. The purpose of this note is to prove some stronger results. We hope that this will stimulate more interest in this problem. Throughout the remainder of this note, we let G be a finite Frobenius group. That is, G contains a subgroup H such that 1 / H / G and

H l H g s 1 for all g g G y H.

A subgroup with these properties is called a Frobenius complement of G. The Frobenius kernel of G, with respect to H, is defined by K s ž/G y D H g j Ä41. ggG

In the language of permutation groups, this corresponds to having a transitive permutation group in which any two-point stabilizer is trivial.

367 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 368 PAUL FLAVELL

The Frobenius kernel corresponds to the set of fixed-point-free elements together with the identity. Clearly, K is a normal subset of G. Character theory is required to prove the following.

THEOREM Ž.Frobenius . K is a subgroup. For the proof seewxwx 4, Theorem 4.5.1, p. 140 or 7, Theorem 5.9, p. 153 . Next we describe the other major results related to the structure of Frobenius groups that we shall need. If K is a subgroup then it is a normal subgroup and every nonidentity element of H induces a fixed-point-free automorphism of K. The following result is the Frobenius Conjecture which was proved by Thompson in his thesis. Seewxw 11᎐13 and 4, Theorem 10.2.1, p. 337x .

THEOREM Ž.Thompson . A finite group that admits a fixed point free automorphism of prime order is nilpotent. The structure of Frobenius complements is also restricted. The following result was obtained by Burnside. Seewx 6, Satz 8.18, p. 506 .

THEOREM Ž.Burnside . Let L be a finite group of fixed point free automor- phisms of a finite group V. Then e¨ery Sylow subgroup of L is cyclic or generalized quaternion. If L has odd order then L is metacyclic. Character theory is not required for the proofs of these results. We mention two further significant achievements.

THEOREM ŽG. Higmanwx 5. . Let V be a finite nilpotent group that admits a fixed point free automorphism of prime order p. Then the nilpotency class of V is bounded by a function that depends only on p.

THEOREM ŽZassenhauswx 9, Theorem 18.6, p. 204. . Let L be a finite insoluble group of fixed point free automorphisms of a finite group V. Then L ( = has a normal subgroup L00 of index at most two such that L SL2Ž.5 M with M metacyclic. Corradi´´ and Horvath show that K is a subgroup provided that G has odd order and NDŽ.­ K for every Sylow subgroup D of G that is contained in K. Without the aid of character theory, we shall prove the following.

THEOREM A. Suppose that G contains a subgroup D / 1 such that D : K, D is not a 2-group, and NŽ. D ­ K. Then K is a subgroup.

COROLLARY B. If G is not simple then K is a subgroup.

COROLLARY C. If G contains a soluble subgroup L such that L l K / 1, L ­ K, and L l K contains an element of odd order then K is a subgroup. NOTE ON FROBENIUS GROUPS 369

Burnside, using the fact that any two involutions generate a soluble subgroup, proved Frobenius’ Theorem in the case that H has even order. Corradi´´ and Horvath extend Burnside’s argument to prove that K is a subgroup provided that H contains an element h of order p, where p s min ␲ Ž.H , such that ²h, h g : is soluble for all g g G. We shall prove the following. ࠻ THEOREM D. Suppose that there exists h g H such that² h, hisg : soluble for all g g G. Then K is a subgroup. The theorem of Frobenius, together with an elementary argumentw 6, Satz 8.17, p. 506x , implies that G has exactly one conjugacy class of Frobenius complements. As far as the author is aware, there is no known character-free proof of this fact. However, a counting argument inspired by Corradi´´ and Horvathwx 2, Lemma 1.2 enables us to prove the following result, which we shall need.

THEOREM E. If L is another Frobenius complement in G then there exists g g G such that H g F LorLg F H.

2. PRELIMINARIES

LEMMA 2.1.Ž. i H is a Hall subgroup of G. Ž.ii <

LEMMA 2.2. Any of the following imply that K is a normal subgroup of G. Ž.i H has e¨en order. Ž.ii H is soluble. Ž.iii K contains a nontri¨ial normal subgroup of G. Ž.iv K contains a nontri¨ial subgroup of G that is normalized by H. Proof. ForŽ. i seewx 9, Prop. 8.3, p. 60 . StatementŽ. ii is the obvious transfer argument, seewx 6, Satz 2.4, p. 417 . Suppose thatŽ. iii or Ž. iv holds. Then there exists a subgroup D / 1 such that D : K and H F NDŽ.. Now H is a Frobenius complement in HD and the kernel of HD has order <<<

3. THE PROOF OF THEOREM E

Assume Theorem E to be false. If H has nontrivial intersection with some conjugate of L then replace L by that conjugate. Let D s H l L. By assumption, H ­ L so D / H. Moreover, if D / 1 then D is a Frobenius complement in H. Let

H˜s ž/H y D D h j Ä41. hgH

Using Lemma 2.1Ž. ii in the case that D / 1, we see that <

<

Dividing through by <

11 1 1 ) yq1 y , <

Recall that D s H l L. It follows that D s H or D s L and then that H F L or L F H, a contradiction.

COROLLARY 3.1. Let L be a subgroup of G such that H l L / 1 and L g H. Then L is a Frobenius group with complement H l L and whose kernel with respect to H l LisKl L. NOTE ON FROBENIUS GROUPS 371

s l Proof. Set H00H L. Then H is a Frobenius complement in L. s y D l j s y D g j Let K0 Ž L l g L H0 . Ä41 . Now K ŽG g g G H . Ä41so l : g x l / x l / K L K0. Let x G be such that H L 1. Now H L L since otherwise H x s H and then L F H, a contradiction. Thus H x l L is also a Frobenius complement in L. Theorem E implies that Ž H x l L. l Ž.H l L lx/ 1 for some l g L. Then H l H l/ 1so H xs H land then xll s xl s l s H L H00. In particular, H K 1. It follows that K L K0.

COROLLARY 3.2Ž. Corradi´´ and Horvath . Any ␲ ŽH .-subgroup of G is conjugate to a subgroup of H.

Proof. Let D / 1bea␲ Ž.H -subgroup of G. Using Sylow’s Theorem and the fact that H is a Hall subgroup of G we may suppose that D l H / 1. Moreover, every element of K has order coprime to <

COROLLARY 3.3. Let P / 1 be a subgroup of G with P : K. Then NPŽ.l K is a normal subgroup of N Ž. P .

Proof. We may suppose that NPŽ.­ K and then that NP Ž.l H / 1. Now apply Corollary 3.1 and Lemma 2.2Ž. iii .

Proof of Theorem D. Let g g G y H and set L s ²h, h g :. Then by Corollary 3.1, H l L is a Frobenius complement in L and the correspond- ing kernel is K l L. By hypothesis, L is soluble so Lemma 2.2Ž. ii implies that K l L is a subgroup and then Lemma 2.1Ž. iv implies that L s ŽH l LK.Žl L .. Also, H g l L is a Frobenius complement in L so by Theorem E there exists l g L such that Ž.H l L l l Ž H g l L.Ž/ 1. Since L s H l LK.Žl L . we may choose l so that l g K l L. Now H lgl H / 1 whence gly1 g H. We deduce that G s HK. Now <<<

4. THE PROOF OF THEOREM A

Throughout this section, we assume the hypothesis of Theorem A. Replacing D by a suitable conjugate, we may suppose that H l NDŽ./ 1. Corollary 3.3 implies that NDŽ.l K is a normal subgroup of ND Ž.. Thus 372 PAUL FLAVELL we may choose a subgroup M F G, maximal subject to Ž.i M l K / 1 and M l H / 1, Ž.ii M l K is a subgroup, and Ž.iii M l K is not a 2-group. Let F s M l K and ␲ s ␲ Ž.F .

LEMMA 4.1.Ž. i F is a nilpotent Hall ␲-subgroup of G. Ž.ii If P / 1 is a characteristic subgroup of F then M s NP Ž ..

Ž.iii F eᎏ M and e¨ery ␲-subgroup of M is contained in F. Proof. Corollary 3.1 implies that M is a Frobenius group with kernel F. By construction, F is a subgroup so Lemma 2.1Ž. iv implies that F is a normal Hall ␲-subgroup of M. This provesŽ. iii . Suppose that P / 1isa characteristic subgroup of F. Then M F NPŽ.. Now NP Ž.l K is a subgroup by Corollary 3.3 so the maximal choice of M forces M s NPŽ.. This provesŽ. ii . Thompson’s Theorem implies that F is nilpotent. Let p g ␲. Since F is ␲ g a nilpotent Hall -subgroup of M we have OppŽ.F Syl ŽM .. Now s g NŽŽ..Op F M by Ž. ii from which it follows that OppŽ.F Syl Ž.G . This provesŽ. i . We recall the following extension of Sylow’s theorem due to Wielandt.

THEOREM 4.2. Let X be a finite group that possesses a nilpotent Hall ␲-subgroup. Then the following hold. Ž.W1 X has a single conjugacy class of Hall ␲-subgroups. Ž.W2 E¨ery ␲-subgroup of X is contained in a Hall ␲-subgroup of X. Ž.W3 If P is a ␲-subgroup of X then N Ž. P possesses a nilpotent Hall ␲-subgroup.

Ž.W4 If Y eᎏ X and T is a Hall ␲-subgroup of X then T l Yisa nilpotent Hall ␲-subgroup of Y. Proof. For W1᎐W3 seew 10, Theorem 3.2, p. 166, and Example 2, p. 186x . W4 follows from the corresponding assertion for Sylow subgroups.

We recall also that a group X is ␲-nilpotent if XrO␲ ЈŽ.X is a ␲-group. If, in addition, X has a Hall ␲-subgroup T then X s TO␲ ЈŽ.X .

LEMMA 4.3. Let L be a subgroup of G with L : K. Then L is ␲-nilpotent. Proof. It suffices to show that L is p-nilpotent for all p g ␲. Choose p g ␲. NOTE ON FROBENIUS GROUPS 373

<<␲ G / g Case Ž.i. 2. Let P 1bea p-subgroup of L. Since OpŽ.F Syl pŽ.G , we may replace L by a suitable conjugate to suppose that F s l P OpŽ.F . Let N NP Ž. K. Note that N is a subgroup by Corollary 3.3 and also that CPŽ.eᎏ N by Lemma 2.1Ž. iii . Choose q g ␲ y Ä4p . Then g OqqŽ.F Syl Ž.G by Lemma 4.1Ž. i and then the Frattini Argument yields s l N CPŽ.Ž. N NŽ.Oq Ž.F . l F s = Using Lemma 4.1Ž.Ž. ii , i we obtain N NŽOqp Ž..F F O Ј Ž.F Op Ž.F s r and hence N CPNŽ.O Ž F . Ž. P. Consequently, N CPŽ.is a p-group. : p r Since L K, we deduce that NPLLŽ.CP Ž.is a p-group for all nontrivial p-subgroups P F L. Frobenius’ Normal p-Complement Theoremw 4, Theo- rem 7.4.5, p. 253x implies that L is p-nilpotent. <<␲ s g / Case Ž.ii . 1. Then F Syl pŽ.G and p 2 since the choice of M g implies that F is not a 2-group. Choose P Syl pŽ.L . We may assume that P / 1 and we shall proceed by induction on <<<

subgroup. Thus NJPLLŽ Ž .. is p-nilpotent. Similarly, NZPŽ Ž .. and hence CZPLŽŽ..are p-nilpotent. Again it follows that L is p-nilpotent.

LEMMA 4.4. Let P be a nontri¨ial subgroup of F. Then ␲ Ž.i NF Ž. P is a Hall -subgroup of N Ž. P , and

Žii .NP Ž .s O␲ Ј ŽNP Ž ..Ž NP Ž .l M .. Proof. We proceed by induction on <

NQŽ.s O␲ Ј Ž.Ž.NQ Ž. NQ Ž.l M .1Ž.

Since P - Q and since CPŽ.: K by Lemma 2.1Ž. iii , we also have

O␲ Ј Ž.NQŽ.F CQ Ž.F NP Ž.l K.2Ž.

It follows fromŽ. 1 and Ž. 2 that

NPŽ.l NQ Ž.s O␲ Ј Ž.ŽNQ Ž. NP Ž.l NQ Ž.l M ..3Ž. 374 PAUL FLAVELL

Using Lemma 4.1Ž. iii we see that NP Ž.l NQ Ž.l F is a Hall ␲-subgroup of NPŽ.l NQ Ž.l M and hence of NP Ž.l NQ Ž.. Recall that Q s s l l ␲ NPF Ž.,soQ NP Ž.NQ Ž. F and thus Q is a Hall -subgroup of NPŽ.l NQ Ž.. Now NP Ž.possesses a nilpotent Hall ␲-subgroup by Theo- rem 4.2Ž. W3 , from which it follows that Q is a Hall ␲-subgroup of NP Ž.. This provesŽ. i .

By Corollary 3.3 we have NPŽ.l K eᎏ NP Ž.so Ž. i and the Frattini Argument yield NPŽ.s Ž.Ž.NP Ž.l KNP Ž.l NQ Ž.. UsingŽ. 3 and Ž. 2 we obtain

NPŽ.s Ž.Ž.NP Ž.l KNP Ž.l NQ Ž.l M s Ž.Ž.NPŽ.l KNP Ž.l M . Now NPŽ.l K is ␲-nilpotent by Lemma 4.3 and Q is a Hall ␲-subgroup of NPŽ.l K. Consequently, NP Ž.l K s O␲ Ј ŽŽ.NP l KQ .. Since O␲ ЈŽŽ.NP l K .F O␲ Ј ŽŽ..NP and Q F NP Ž.l M we deduce that NPŽ. s O␲ ЈŽNP Ž ..Ž NP Ž .l M ., which provesŽ. ii .

LEMMA 4.5. Any two elements of F that are conjugate in G are already conjugate in M. Proof. Let f g F and g g G be such that f g g F. Now ZFŽ.g F CfŽ g . g and Lemma 4.3, together with Theorem 4.2Ž. W4 , imply that CfF Ž. is a nilpotent Hall ␲-subgroup of CfŽ g .Ž. Thus there exists c g Cfg . such that ZFŽ.gc F F. It suffices to show that gc g M. Now NZFŽŽ.gc. s M gc and hence F gc is the unique Hall ␲-subgroup of NZFŽŽ.gc.. Moreover, gc ␲ gc NZFF ŽŽ..Žis a Hall -subgroup of NZFŽ..Žby Lemma 4.4 i. . We deduce that F gc s F and then that gc g NFŽ.s M, as desired.

LEMMA 4.6. H F M.

Proof. Let Ä4f1,..., f␣ be a set of representatives for the conjugacy classes of nontrivial ␲-elements of G. By Theorem 4.2Ž. W2 and W1 we g may suppose that fi F for all i. For each i let s Tiif O␲ Ј Ž.CfŽ.i , : : and note that TiiCfŽ. K. Recall that each element of G can be expressed uniquely as a commuting product of a ␲-element and a ␲ Ј-ele- ment. It follows that for all i, j and all g g G that

l g / л s g TijT if and only if i j and g CfŽ.i. NOTE ON FROBENIUS GROUPS 375

Thus ␣ <<) << K Ý G : CfŽ.ii T. is1 s It follows from Lemma 4.4 that CfŽ.i O␲ Ј ŽŽ..Ž.CfiFi C f for each i. Moreover, <

Lemma 4.5 implies that Ä4f1,..., f␣ is a set of representatives for the conjugacy classes of nontrivial ␲-elements of M. By Lemma 4.1Ž. iii , the set of ␲-elements of M is F whence <

Proof of Corollary B. Assume false and choose Leᎏ G such that 1 / L / G. Then L ­ K by Lemma 2.2Ž. iv so as Leᎏ G we deduce that L l H / 1. Now L l H / L since H cannot contain a nontrivial normal sub- group of G. Thus L l H is a Frobenius complement in L so Corollary 3.2 and the Frattini Argument imply that G s LNŽ. L l H . Consequently, G s LH.4Ž. << g Let p be any prime divisor of G : H and choose P Syl pŽ.G . Since H is a Hall subgroup of G we have P : K and usingŽ. 4 we also have P F L. Thus G s LNŽ. P and then s <<

Now <

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