Group algebras in which the socle of the center is an ideal (Q6580059)

Let \(F\) be a field of prime characteristic \(p>0\) and let \(G\) be a finite group and let \(ZFG\) denote the center of the group algebra \(FG\). \N\NFinite groups \(G\) such that the Jacobson radical \(J(ZFG)\) is an ideal of \(FG\) are characterised by \textit{R. J. Clarke} [J. Lond. Math. Soc., II. Ser. 1, 565--572 (1969; Zbl 0206.31502)], \textit{S. Koshitani} [J. Lond. Math. Soc., II. Ser. 18, 243--246 (1978; Zbl 0389.20003)] and \textit{B. Külshammer} [Arch. Math. 114, No. 6, 619--629 (2020; Zbl 1457.20004)]. In the article under review, the authors consider the corresponding problem for the socle of \(ZFG\) and for the Reynolds ideal of \(FG\). \N\NTheir first main result establishes that the Reynolds ideal of \(FG\) is an ideal in \(FG\) if and only if the derived subgroup \(G'\) of \(G\) is contained in the \(p\)-core \(O_p(G)\) of \(G\). \N\NNow, provided that the socle of \(ZFG\) is an ideal in \(FG\), the latter result implies that the group \(G\) admits a decomposition as a semi-direct product \(G\cong P\rtimes H\) where \(P\) is a Sylow \(p\)-subgroup of \(G\) and \(H\) is an abelian \(p'\)-group. This decomposition is then used to characterise all \(p\)-groups \(G\) such that the socle of \(ZFG\) is an ideal in \(FG\). This is the case if and only if the nilpotency class of is at most two, or \(p = 2\) and \(G'\subseteq Y(G)Z(G)\) where \(Y(G) =\langle fg-1 \mid \{f, g\}\) is a conjugacy class of length \(2\) of \(G \rangle\). \N\NBack to arbitrary finite groups, generalising the \(p\)-group case, the authors prove further criteria for the socle of \(ZFG\) to be an ideal in \(FG\) leading them to the following group-theoretic characterisation, which is their second main result: if \(G\) is a finite group for which \(\mathrm{soc}(ZFG)\) is an ideal in \(FG\), then \(G\) is the central product of the centraliser \(C_P(H)\) and the \(p\)-residual group \(O^p(G)\).