On an <i>IC</i> -property of subgroups of a finite group (Q6636319)

Let \( H \) be a subgroup of a finite group \( G \) and let \(p\) be a prime. \( H \) is called an \(IC_{p}\)-subgroup of \(G\) if\N\begin{itemize}\N\item[(i)] \(H\cap [H,G]\) contains \(H_{G}\), where \(H_{G}\) denotes the largest normal subgroup of \(G\) contained in \(H\),\N\item[(ii)] the factor group \(H\cap [H,G]/H_{G}\) is a \(p'\)-group.\N\end{itemize}\N\NThe authors investigate this definition and prove several results showing that, under specific assumptions about various subgroups of a finite group, certain conditions imply solvability.