On the Fitting height of a soluble group that is generated by a conjugacy class of 3-elements

Abstract

Let G be a finite soluble group that is generated by a conjugacy class consisting of elements of order 3. We show that there exist four conjugates of an element of order 3 that generate a subgroup with the same Fitting height as G. We use this result to find a soluble analogue of the Baer–Suzuki theorem in the case prime 3.

Multilinear commutators in residually finite groups

“…Then dim C V (a) ≤ 3 4 dim V . The particular case of the above lemma where a has order 3 was established earlier in Al-Roqi and Flavell [3]. This enabled the authors to prove that if G is a soluble group containing an element a of order 3 such that G = a G , then there exist four conjugates of a that generate a subgroup with the same Fitting height as G. Using Lemma 2.2 in place of the particular case of 3-elements in the Al-Roqi and Flavell arguments we see that the assumption that a is of order 3 can be dropped and so we obtain the following lemma.…”

On the Fitting height of a soluble group that is generated by a conjugacy class of 3-elements (II)

“…We state the definitions of some nonstandard symbols from [1]. Let G be a finite group, P a subgroup of G with prime order p and n a natural number.…”

On representations of groups of odd order

“…The properties of the soluble groups generated by four conjugates of some element have been discussed in [1] and the case of soluble groups generated by three conjugates has been discussed in [2]. Using Theorem 1.1, we derive some properties of soluble groups of odd order that are generated by two conjugates of an element of prime order.…”

“…Let G be a finite soluble group of odd order generated by the conjugacy class of a cyclic group P of prime order. If V is a faithful irreducible G-module over a field then dim C V ðPÞ < The proof uses results on the Fitting height of a soluble group by Flavell in [4] and by Al-Roqi and Flavell in [1], [2], [3].…”

“…We need a generalization of [1,Theorem 3.3], replacing the prime 3 by an odd prime. Although the proof runs parallel to the original one, for the sake of completeness, we present a complete proof.…”