\(C\)-normality of groups and its properties (Q1911796)

A subgroup \(H\) of finite group \(G\) is called \(c\)-normal in \(G\) if there exists a normal subgroup \(N\) of \(G\) such that \(G=HN\) and \(H\cap N\leq H_G\), the normal core of \(H\) in \(G\). The author derives several results relating the structure of \(G\) and its \(c\)-normal maximal subgroups. Theorem 3.1. \(G\) is solvable if and only if every maximal subgroup of \(G\) is \(c\)-normal in \(G\). Theorem 3.2. A maximal subgroup \(M\) of \(G\) is \(c\)-normal in \(G\) if and only if \(\eta(G:M)\), the normal index of \(M\) in \(G\), \(=[G:M]\). Theorem 4.1. \(G\) is supersolvable if \(P_1\) is \(c\)-normal in \(G\) for every maximal subgroup \(P_1\) of each Sylow subgroup \(P\) of \(G\).