Carter subgroups and Fitting heights of finite solvable groups

Abstract

Let G be a finite solvable group having Fitting height h (as defined in [7] or in 1 below).Let H be a Carter subgroup of G and be the length of a composition series of H.We shall establish the correctness of a conjecture of John Thompson (at the end of [7] by proving that (0.1) h_< 10(2 -1) 41.This is the result of Theorem 8.5 below, and the rest of this paper is a proof of that theorem.The Fitting series Fn(G), n 0, 1, 2, is defined inductively by Fo(G)then here is some integer h >_ 0 such that F(G) G.We call the least such integer h the Fitting height of G and denote it by h(G).If each S, i 1, ], is an element or a subset of G, then (S, S) will denote the subgroup of G generated by S, S.If , r e G, then we define CTr --I T 0"T 0" T 0" T (TT 0" 0".CARTER SUBGROUPS AND FITTING HEIGHTS zi5l For any rl, r e G and any integers al, am, we define alr ...q--an- O" (o'al)Vl(o'a2) v2(o'an)TM, for all -1+ for all, reG.Ifp,The third basic theorem, whose proof will follow later (see 7) is THEOREM 2.13.Let H be a group acting on an augmented Fitting chain A1, A, {Bi}.Suppose that P is a normal subgroup of order 3 in H such that [A1, P] /1}.If >_ 6, then there is an H-invariant augmented Fitting subchain D6 D {C} of A6 At, {B} such that P centralizes each Dj and Ci Assuming the three Theorems 2.6, 2.7 and 2.13, we now prove the following result from which we shall later derive a proof of Thomson's conjecture (see 8).THEOREM 2.14.Let a finite group H act on an augmented Fitting chain A i, At, {Bi} 80 that H centralizes no non-trivial section of any Ai, j 1, t. Assume further that H is a supersolvable group with a normal 3- Sylow subgroup M. Then _ 5(2 (") 1).Proof.We use induction on M [.If M[ 1, then H and A1, At satisfy the hypotheses of Theorem 2.8.That theorem tells us thatt_ 3(2 (s) 1) _ 5(2 (s) 1).So this theorem is true iflM[ 1.Now we assume that M ) 1 and that this theorem is true for all smaller values of M I. Since H is supersolvable it has a normal subgroup P of prime order p.We may even choose P to be contained in the normal 3-Sylow sub- groupMofH.Sop 3.Suppose that P centralizes A1,As, for some integer s 1, t.