The influence of minimal subgroups of focal subgroups on the structure of finite groups (Q1602682)

A subgroup \(H\) of a group \(G\) is said to be complemented in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K=1\). The authors prove that if \(p\) is a prime dividing the order of a group \(G\) and \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(G\) is \(p\)-nilpotent whenever every minimal subgroup of the focal subgroup \(P\cap G'\) is complemented in \(N_G(P)\) and \(N_G(P)\) is \(p\)-nilpotent (Theorem~2.1). When \(p\) is the smallest prime dividing the order of \(G\), the condition ``\(N_G(P)\) is \(p\)-nilpotent'' can be removed (Theorem~2.2). Some applications of these results are shown, for example, if every minimal subgroup of \(P\cap G'\) is complemented in \(N_G(P)\) for every Sylow subgroup \(P\) of \(G\), then \(G\) has a Sylow tower of supersoluble type (Corollary~2.3), or if \(\mathfrak F\) is a formation containing the class \(\mathfrak U\) of supersoluble groups and \(H\) is a normal subgroup of a group \(G\) such that \(G/H\in{\mathfrak F}\), and for every Sylow subgroup \(P\) of \(H\), every minimal subgroup of \(P\cap G'\) is complemented in \(N_G(P)\), then \(G\in{\mathfrak F}\) (Theorem~2.4). As a consequence, a group \(G\) is supersoluble provided that for every Sylow subgroup \(P\) of \(G'\), every minimal subgroup of \(P\) is complemented in \(N_G(P)\). If \(G\) is a soluble group, the thesis of Theorem~2.4 can be achieved by considering only the minimal subgroups of the Fitting subgroup \(F(G'\cap H)\) of \(G'\cap H\). As a consequence, if \(G\) is a soluble group and every minimal subgroup of the Fitting subgroup \(F(G')\) of \(G'\) has a complement in \(G\), then \(G\) is supersoluble (Corollary~2.9). In Theorem~2.10, it is proved that if \(\mathfrak F\) is a saturated formation containing the class \(\mathfrak U\) of supersoluble groups and \(H\) is a normal subgroup of a group \(G\) such that \(G/H\in{\mathfrak F}\), if all maximal subgroups of every Sylow subgroup of \(H\) are complemented in \(G\), then \(G\in{\mathfrak F}\).