Date: Tue, 10 Dec 1996 08:32:07 +1100
From: CASOLO@udini.math.unifi.it

Marty Isaacs asked
 
> 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?
 
It seems to me that it might be seen as a particular
case of theorems concerning the solvable residual of
a join of subnormal subgroups (the most general ones
are due to J. Roseblade), here applied to the subnormal
subgroups  G  and  KA  in the semidirect product  GA.
In a sense, it was known to Wielandt: he proved that
if the finite group  W  is generated by subnormal
subgroups  H  and  K  then the soluble residual of  W
is the product of the soluble residuals of  H  and  K
(cfr. Lennox and Stonehewer Subnormal Subgroups
of Groups, Th. 4.4.1).
 
best wishes
carlo casolo