Automorphisms of soluble groups (Q2809273)

The paper under review considers only finite groups \(G\). Let \(r\) be a prime and let \(R\) be a group of order \(r\) acting on a finite \(r'\)-group \(G\) such that \([G, R]\) is soluble. As usual, if \(F\) is a field, \(F[G]\) denotes the group algebra of \(G\) over \(F\). If \(H\) is a subgroup of \(G\) and if \(V\) is a \(F[G]\)-module, then \(V_H\) denotes \(V\) considered as an \(F[H]\)-module.NEWLINENEWLINEThe author considers the situation where \(V\) is an \(RG\)-module, possibly of mixed characteristic (i.e. a direct sum of \(F[RG]\)-modules, where the field \(F\) can vary for the summands), with \(V_{[G, R]}\) faithful and completely reducible. Since \(C_V(R)\) is a \(C_G(R)\)-module, a normal subgroup \(K\) of \(C_G(R)\), which is also contained in the kernel of \(C_G(R)\) on \(C_V(R)\), has an unknown subnormal closure \(L\) in \(G\) which the author sets to determine under the above hypotheses.NEWLINENEWLINEIn the main result, it is shown that \(L=K[L, R]\) and that \([L, R]\) is nilpotent. Moreover, \(V\) decomposes as the direct sum of three \(RL\)-submodules whose internal structure is described in more technical detail.NEWLINENEWLINEPrevious results on coprime action on solvable groups of \textit{G. Glauberman} [Can. J. Math. 20, 1465--1488 (1968; Zbl 0167.02602)] and \textit{J. G. Thompson} [J. Algebra 1, 259--267 (1964; Zbl 0123.02602)] are obtained as corollaries. Corollary E is less technical and can be stated here:NEWLINENEWLINE Let \(R\) be a group of prime order \(r\) that acts on the soluble \(r'\)-group \(G\). Let \(p\) be a prime and let \(P\leqslant O_p(C_G(R))\). Assume that \([C_G(P), R]=1\). Then \([G, R]\leqslant O_{p'}(F(G))\).NEWLINENEWLINE The conclusion of Corollary E is very powerful, it shows \([G, R]\) to be rather small under certain conditions.NEWLINENEWLINEThe paper is written with great care for the reader and it contains useful sections with background on groups acting on groups, on modules, on primitive modules and on Hall-Higman theory. It is in fact surprising that the investigation of the subnormal closure in \(G\) of a subgroup of the fixed-point subgroup of an automorphism of \(G\) could lead to such deep results.