\(E\)-constrained groups. (Q1942317)

Let \(G\) be a finite group. We say that \(G\) is a quasisimple group if \(G'=G\) and \(G/Z(G)\) is simple. The subgroup \(E(G)\) of \(G\) generated by all subnormal quasisimple subgroups (components) of \(G\) is called the layer of \(G\). The subgroup \(F^*(G)=E(G)F(G)\), where \(F(G)\) is the Fitting subgroup of \(G\), is called the generalized Fitting subgroup of \(G\). It is well known that \(C_G(F(G))\leq F(G)\) for all finite soluble groups \(G\) and \(C_G(F^*(G))\leq F^*(G)\) for all finite groups \(G\). \textit{P. Flavell} and \textit{J. Medina} [Arch. Math. 82, No. 1, 1-3 (2004; Zbl 1060.20017)] showed that \(C_G(F(G))\leq F(G)\) if and only if \(G\) contains a nilpotent subgroup \(I\) satisfying \(C_G(I\cap I^g)\leq I\cap I^g\) for all \(g\in G\). A group \(G\) is called \(E\)-constrained if \(C_G(E(G))\leq E(G)\), i.e. \(F^*(G)=E(G)\). In this paper, the authors prove a similar theorem for the \(E\)-constrained groups: \(C_G(E(G))\leq E(G)\) if and only if \(G\) contains a subgroup \(K\) such that \(K'=K\) and \(K/Z(K)\) is a direct product of simple groups, and \(C_G((K\cap K^g)')\leq K\cap K^g\) for all \(g\in G\). The classification of finite simple groups is used for the proof of the statement of sufficiency of this theorem.