Finite solvable groups with \(C\)-closed invariant subgroups. (Q2473760)

A subgroup \(H\) of a finite group \(G\) is defined as \(C\)-closed if the centraliser of the centraliser of \(H\) in \(G\) is \(H\) itself. The concept goes back to \textit{W. Gaschütz} [Arch. Math. 6, 5-8 (1954; Zbl 0057.02002)], who characterised those finite groups all subgroups of which have this property. In the present paper, finite soluble groups are considered all normal subgroups of which are \(C\)-closed (Theorem 1): If \(G\) is such a group, it can be represented as the direct product of subgroups \(G_i\), where (1) each \(G_i\) is a semidirect product of \(P_i\) by \(\langle x_i\rangle\), \(P_i\) being the commutator subgroup of \(G_i\) and of prime-power order while the orders of the elements \(x_i\) are square-free and pairwise relatively prime, and, (2) if \(o(x_i)=q_1q_2\cdots q_n\), and each \(q_i\) is a prime, \(G_i\) has precisely \(n\) minimal normal subgroups \(A_j\) with the property that \(C_{G_i}(P_i)=Z(P_i)=\prod^n_{j=1}A_j\) such that, for \(j= 1,2,\dots,n\), \(C_{G_i}(A_j)\) is a semidirect product of \(P_i\) by \(\langle x^{q_j}_i\rangle\), numbering the factors.