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