On partial cap-subgroups of finite groups (Q2285301)

This paper deals with finite groups \(G\) in which for some prime power \(p^d\) such that \(1<p^d<|P|\), where \(P\) is a Sylow \(p\)-subgroup of \(G\), all subgroups of order \(p^d\) and, when \(p^d=2\) and \(P\) is nonabelian, all cyclic groups of order~\(4\) are partial CAP-subgroups of \(G\), that is, for each such subgroup \(A\) there is a chief series of~\(G\) whose associated chief factors are either covered or avoided by~\(A\). These groups are said to satisfy the hypothesis \(\operatorname{HY}(p^d)\). These groups are known to be \(p\)-soluble with \(p\)-length~\(1\). The aim of this paper is to precise the structure of groups with \(\operatorname{HY}(p^d)\) and \(p\)-rank greater than~\(1\). For groups with \(\operatorname{HY}(p^d)\) and \(\operatorname{O}_{p'}(G)=1\) and \(r_p(G)\ne 1\), it is proved (Theorem~A\('\)) that \(G=H \ltimes P\), where \(H\in\operatorname{Hall}_{p'}(G)\) and \(P\in\operatorname{Syl}_p(G)\); furthermore, let \(V\) be an irreducible \(\mathbb{F}_p[H]\)-submodule of \(P/\Phi(P)\) and write \[\dim_{\mathbb{F}_p} V=e,\quad d'=d-\log_p|\Phi(P)|,\quad n'=\log_p|P/\Phi(P)|,\] then the following statements hold: (1) \(P/\Phi(P)\) is a homogeneous \(\mathbb{F}_p[H]\)-module, while \(V\) is not absolutely irreducible; (2) \(d'\ge e\ge 2\), \(e| \gcd(d',n')\), also \(G/\Phi(P)\) satisfies \(\operatorname{HY}(p^e)\); (3) \(H\) is supersoluble and its Sylow subgroups are all abelian, moreover, the Fitting subgroup of \(H\) is cyclic. The case \(\operatorname{HY}(p^2)\) is studied in Theorem~B\('\): Let \(G\) be not \(p\)-supersoluble with \(\operatorname{O}_{p'}(G)=1\) and \(|G|_p\ge p^3\). Then \(G\) satisfies \(\operatorname{HY}(p^2)\) if and only if the following statements hold: (1) \(G=H\ltimes P\), where \(H\in\operatorname{Hall}_{p'}(G)\) and \(P\in\operatorname{Syl}_p(G)\); (2) \(H\) is cyclic; (3) \(P=V_1\times \dots \times V_r\), \(r\ge 2\), where the \(V_i\) are \(H\)-isomorphic irreducible \(\mathbb{F}_p[H]\)-modules of dimension~\(2\). The case of \(2\)-maximal subgroups is analysed in Theorem~C\('\): Let \(G\) be a non-\(p\)-supersoluble group with \(\operatorname{O}_{p'}(G)=1\), and let \(P\in\operatorname{Syl}_p(G)\) with \(|P|\ge p^3\). Then all \(2\)-maximal subgroups of \(P\) are partial CAP-subgroups of~\(G\) if and only if \(G\) is of one of the following types: (1) \(G=H\ltimes P\), where \(H\in\operatorname{Hall}_{p'}(G)\) is cyclic, \(\Phi(P)\) coincides with the intersection of all \(2\)-maximal subgroups of~\(P\), \(P/\Phi(P)=V_1\times\dots\times V_r\), \(r\ge 2\), where the \(V_i\) are \(H\)-isomorphic irreducible \(\mathbb{F}_p[H]\)-modules of dimension~\(2\); or (2) \(p=2\) and \(P\cong Q_8\). In particular, it is obtained that \(G/{\operatorname{O}_{p',p}(G)}\) is a supersoluble group. These results precise the ones obtained in [\textit{A. Ballester-Bolinches} et al., J. Alg. 342, 134--146 (2011; Zbl 1243.20030); Sci. China Math. 55, 961--966 (2012; Zbl 1264.20018); J. Pure Appl. Algebra 215, 705--714 (2011; Zbl 1223.20012)].