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The Fitting Length of a Finite Soluble Group and the Number of Conjugacy Classes of Its Maximal Nilpotent Subgroups

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The Fitting Length of a Finite Soluble Group and the Number of Conjugacy Classes of Its Maximal Nilpotent Subgroups BULL. AUSTRAL. MATH. SOC. MOS 20D25, 20D30 VOL. 6 (1972), 213-226. (20DI5) The Fitting length of a finite soluble group and the number of conjugacy classes of its maximal nilpotent subgroups A.R. Makan It is shown that there exists a logarithmic upper bound on the Fitting length h(G) of a finite soluble group G in terms of the number v(G) of the conjugacy classes of its maximal nilpotent subgroups. For v(G) = 3 , the best possible bound on h{G) is shown to be h . 1. Introduction All groups considered in this paper are finite and soluble, and, for e group G , v(G) denotes the number of conjugacy classes of its maximal nilpotent subgroups and h(G) denotes its Fitting length. In [5], it was shown that the Fitting length of a finite soluble group of odd order is bounded in terms of the number of conjugacy classes of its maximal nilpotent subgroups. Our main purpose of this paper is to show that the result is true for any finite soluble group, not necessarily of odd order. More precisely, we show here that THEOREM 1.1. For any finite soluble group G , Received 19 October 1971- This work is part of the author's PhD thesis at the Australian National University. The author thanks Dr L.G. Kovacs and Dr H. Lausch for their invaluable guidance and constant encouragement during the preparation of this work. He also thanks Dr M.F. Newman for allowing him access to his unpublished result used here and for many helpful conversations, and Dr T.M. Gagen for his many helpful comments. 213 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 25 Sep 2021 at 09:10:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700044440 214 A.R. Makan This result improves considerably the bound obtained in [5] for a finite group of odd order. The precise form of the bound obtained (though not its order of magnitude) relies, however, on an unpublished result of Newman (see Theorem 3.5)- Theorem 1.1 is proved in Section 3. In Section 4, we obtain the best possible bound on h(G) in the special case when v(G) = 3 . The sharp bound on h{G) in the case when v(G) =2 or in the case when v(G) = 3 but |G| is odd has been discussed in [5]. 2. Notation We will use the following notation in the rest of the paper: = the order of the group G Z(G) = the centre of G NG(H ') = the normalizer in G of the subgroup H of G CG(H') = the centralizer of H in G *(G) = the Frattini subgroup of G F{G) = the Fitting subgroup of G 0 i (6) = the largest normal p-nilpotent subgroup of G V V Op,(G) = the largest normal p'-subgroup of G d(G) = the minimal number of generators of G H < G reads H is a subgroup of G H < G reads H is a proper subgroup of G H 5 G reads H is a normal subgroup of G H o G reads H is a proper normal subgroup of G G x = the direct product of the groups G and H p, q will always denote primes GF(p) = the field with p elements. 3. The proof of the theorem First, we will prove some lemmas. We begin with the following Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 25 Sep 2021 at 09:10:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700044440 Fitting length and conjugacy classes 21s elementary lemma. LEMMA 3.1. Let G be a group whose Fitting subgroup F is a p-group and let Q be a q-subgroup of G , q + p . Then there is a maximal nilpotent subgroup W of G such that CAQ) = W n F . Moreover, if Q t (I) , then W n F < F . Proof. Let W be a maximal nilpotent subgroup of G which contains CAQ) x © . Clearly W n F = CAQ) . Also, since CJF) < F (see Theorem r t & 6.1.3 in Gorenstein [3]), F £ W unless © = {l} . Thus, if © # {l} , W n F < F , as required. Next, we show LEMMA 3.2. Let G be a group whose Fitting subgroup F is an elementary abelian p-group, let Q be a non-trivial q-subgroup of G , q # p j and let S be a maximal element in the set X = {Q* I Q* o Q, Cp(Ql < Cp(Q*)} . Then CQ[CF{S)) = S and, moreover, Z(Q/S) is cyclic. Proof. Since S 2 Q , C_(5) is ^-invariant. Thus H = CQ{CF(SJ) o « . However, S5S and also Cp{Q) < C?(S) S Cf(ff) . Hence, since S is a maximal element in X, , it follows that 5 = # , as required. In order to show that Z{Q/S) is cyclic, we proceed as follows. Since Cp{S) is ©-invariant, we observe first that, by Theorem 3.3.2 in x Gorenstein [3], C_(5) = Cp(Q) L , where L is ©-invariant. Clearly, L * {l} since CAQ) < CAS) . Now, let L* be a non-trivial ©-invariant subgroup of L of minimal order and let K = CAL*) . Since L* is ©-invariant, K < © . Moreover, K S CQ[L*C„(©)) . In particular, CF(©) < Cp{K) , and so, K t X . However, S S K . Thus, since S is a maximal element of ^ , X = S , and hence ©/S is represented faithfully as well as irreducibly on L* , regarded as a vector space over the field GF(p) . By Theorem 3-2.2 in Gorenstein [3], it follows then that Z(©/5) is cyclic, and the proof is complete. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 25 Sep 2021 at 09:10:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700044440 216 A.R. Makan The following lemma extends Lemma 3 of [5D. LEMMA 3.3. Let G, F and Q be as in Lemma 3.2 and let I be the largest integer for which there exists a chain of subgroups (3.10 Cp(Q) = Vt n F < VU1 n F < ... < V± n F < F , where V. is a maximal nilpotent subgroup of G for i = 1, 2, ..., I and Cp(Q) x Q £ V . If Q/Z(Q) is elementary abelian, then d{Q) is at most 2l . Proof. We proceed by induction on \Q\ . Let 3C be defined as in Lennna 3.2, and let R be a maximal element in X . Then, by the same lemma, Z/R = Z(Q/R) is cyclic. Also, since Cp(Q) < Cp(R) and, by Lemma 3.1, there is a maximal nilpotent subgroup W of G such that W n F = Cp(R) , it follows, in view of our hypothesis, that a chain of subgroups of the type (3.U) which joins CAR) to F has length at most I - 1 . Suppose first that Q = Z , so that Q/R is cyclic. Then, since, by induction, d{R) 5 2{l-l) , d(Q) 5 2.1 - 1 and we are done. Hence, assume next that Q t Z . Let A/R be a maximal abelian normal subgroup of Q/R . Since Q/Z(Q) , and therefore Q/Z is elementary abelian, it follows from Satz III, 13-7 in Huppert [4] that there is a maximal abelian normal subgroup B/R of Q/R such that AB = Q , A r> B = Z and d(A/Z) = d{B/Z) ; consequently d(Q/R) £ 2d(A/R) . It remains now to show that d(A/R) 5 I and that a chain of subgroups of the type (3-1*) which joins Cp{R) to F has length at most I - d(A/R) , for then, by the inductive hypothesis, d(R) £ 21 - 2d{A/R) £ 21 - d(Q/R), and hence d(Q) £ d{R) + d(Q/R) £ 21 . We show this as follows. Let A = A and for i = 1, 2, ... , define A . to be a maximal element in the set {Q* | R £ Q* £ Ai_1 and C^A^ < Cp(Q*)} . For some integer n £ 1 , 4 = i? . Now, let G be the semidirect product of X = CJi?) by y = 4/i? . Since, by Lemma 3.2, CV(X) = {l} , X is Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 25 Sep 2021 at 09:10:04, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700044440 Fitting length and conjugacy classes 217 clearly the Fitting subgroup of G . Thus, by the same lemma and the fact that A/R is abelian, it follows now that A . I A . is cyclic for i = 1, 2, ..., n . In particular, d{A/R) £ n . On the other hand, by Lemma 3-1, there exist maximal nilpotent subgroups W, W , ..., W of G such that C = W n F < C = W n F < < C A = C {R) W n F F^ 0 F^ l ••• F^ r) F = n • Hence, by our hypothesis, n £ I , whence d{A/R) £ n S I . Also, for the same reason, the chain of subgroups of the type (3-^) joining C_(i?) to F has length certainly at most I - n 5 I - d(A/R) .
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