More of a question than a problem really, but so what? GCS

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Date: Tue, 03 Dec 1996 15:53:35 -0600
From: isaacs@math.wisc.edu
Subject: repeated commutators
 
Let A act on G where A and G are finite and let
[G,A,A, ... ,A] = K. Finally, let B be the largest subgroup
of A such that [G,B] is contained in K. (Thus B is the
kernel of the action of A on the set of right cosets of
K in G.)
 
I found some old notes in my handwriting that prove
(among other things) that if K contains some term of
the derived series of G then A/B is solvable. I do not
know from where I got this result. It is even possible
that I discovered it. My question is: does anyone have
a reference for this; is it actually a known result?
 
Remark: If K is normal in G, it is no loss to assume that
K = 1 and in that case, A/B is actually nilpotent by a result
of P. Hall. (Without any solvability assumption on G.) My
interest, therefore, is specifically in the case where
K is not normal, and is, in fact core-free. 
 
I. M. Isaacs
isaacs@math.wisc.edu