Character Theory and Length Problems

A Fitting height lemma and its applications

“…Let A = ϕ be a cyclic group acting on a solvable group G. In that case, the problem of finding a bound for h(G) in terms of |A| and |C G (A)| is still open (see the Kourovka Notebook [7, 13.8(a)]). However, there are some nice bounds proved in the case that the action is fixed point free, that is, when C G (A)=1: Under some further assumptions on the structure of A and G, a linear bound has been obtained for h(G) depending on l(A) in [2,3], and [8]. A current article established a quadratic bound h(G) ≤ 7l(A) 2 without any further assumptions on A or G (see [6,Corollary 1.2]).…”

Bounding the fitting height in terms of the exponent

“…Quite a few investigations have been devoted to finding relations involving the Fitting height and other invariants. The interested reader may get an idea of such kind of problems by consulting the survey [5] and its bibliography. F 0 (G) = 1 and F i (G)∕F i−1 (G) = F(G∕F i−1 (G)) for every i ≥ 1 .…”

About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups

“…[21], Alexandre Turell ( see https://people.clas.ufl.edu/turull/ ) states a conjecture of Thomas R.Berger (which actually dates back to John G. Thompson in the 1970's):Conjecture 2.3 ( see[2] ) Let p be a prime. There exists a linear function f p such that if G is a finite p-soluble group with p-length λ p(G) and P is a subgroup of G of order p k ( k  ℕ )contained in precisely one Sylow p-subgroup then λ p(G)  f p (k ) .Having studied most of the hereof related literature published by Brian Hartley, Andrew Rae, and Thomas R. Berger, we profess to have discovered such a linear function, namely our ap .Therefore we can state Berger's conjecture more precisely ( and best possible ) as Let p be a prime.…”