The influence of \(p\)-regular class sizes on normal subgroups. (Q2869866)

The main goal of the paper is to improve a result on the same subjects as shown in [J. Algebra 336, No. 1, 236-241 (2011; Zbl 1241.20034)], due to the same three authors and \textit{M. Khatami}. As such, the reader will observe now in the paper under review, the following result, thereby eliminating prescribing the \(p\)-solvability of the given group \(G\).NEWLINENEWLINE Theorem A. Let \(N\) be a normal subgroup of a finite group \(G\). Let \(p\) be a prime number and suppose that the \(G\)-conjugacy class of every \(p\)-regular element of \(N\) has size 1 or \(m\) for some fixed integer \(m\). Then \(N\) has abelian \(p\)-complements or \(N=RP\times A\), where \(R\) and \(P\) are a Sylow \(r\)-subgroup for some prime \(r\) and Sylow \(p\)-subgroup of \(N\), respectively, and \(A\) is a central subgroup of \(G\).NEWLINENEWLINE Theorem A provides also an extension of the main result of \textit{E. Alemany, A. Beltrán} and \textit{M. J. Felipe} [Proc. Am. Math. Soc. 139, No. 8, 2663-2669 (2011; Zbl 1236.20036)].NEWLINENEWLINE One of the elucidating parts of the proof of Theorem A is the knowledge of the Schur multiplier of some simple groups.