Groups satisfying a strong complement property (Q2314986)

Let \(G\) be a finite group and \(N\) a normal subgroup of \(G\). A complement for \(N\) in \(G\) is a subgroup \(H\) such that \(G=NH\) and \(N\cap H=1\). Let \((h_{1},h_{2}, \ldots, h_{d})\) be a generating sequence of \(H\) (i.e. \(h_{1},h_{2}, \dots, h_{d}\) is a set of generators for \(H\)), then \((N,H)\) satisfies the strong complement property if \(\langle h_{1}^{x_{1}}, h_{2}^{x_{2}}, \ldots, h_{d}^{x_{d}} \rangle\) is a complement of \(N\) in \(G\) for all \(x_{1},x_{2}, \ldots, x_{d} \in N\). Let \(d(H)\) be the minimal number of elements needed to generate \(H\). It is easy to see that if \(d > d(H)\) and \((N,H)\) satisfies the strong complement property for any generating sequence \((h_{1},h_{2}, \ldots, h_{d})\) of \(H\) with length \(d\), then \(H\) acts trivially on \(N\). When \(d=d(H)\) and \((N,H)\) satisfies the strong complement property for any generating sequence \((h_{1},h_{2}, \ldots, h_{d})\) of \(H\), then \((N, H)\) satisfies the strong complement property. In this paper, it is proved that if \((N,H)\) satisfies the strong complement property with \((\vert N\vert,\vert H \vert)=1\), then \(H\) is cyclic or \(H\) acts trivially on \(N\). In the case when \(N\) and \(H\) are not of coprime order, the authors provide examples where \((N,H)\) satisfies the strong complement property and where \(H\) is not cyclic and does not act trivially on \(N\).