The vertices of the components of the permutation module induced from parabolic groups (Q1626402)

Let \(G\) be a finite group with subgroup \(H\), \(k\) be an algebraically closed field of characteristic \(p\). A \(p\)-subgroup \(V\) of \(G\) is called \(H\)-suitable (with respect to \(G\) and \(p\)), if for every \(S\in \mathrm{Syl}_p(G)\) with \(V \leq S\) there exists some \(g\in G\) so that \(V = S \cap H^g\), where \(H^g = g^{-1}Hg\). It is known [the author, Commun. Algebra 43, No. 2, 659--673 (2015; Zbl 1314.20006)] that \(H\)-suitability is a necessary condition for a \(p\)-group to be the vertex of a component of the permutation module \(k_H\!\uparrow ^G\). Let \(G = \mathrm{GL}_n(q)\) be the general linear group, where \(q = p^f\). Let \(\lambda = (\lambda_1,\dots,\lambda_r)\) be a composition of \(n\), that is, \(\lambda_1, \dots, \lambda_r\) are positive integers such that \(\lambda_1 + \cdots +\lambda_r =n\). Let \(P_{\lambda}\) be a parabolic subgroup of \(G\) consisting of matrices from \(\mathrm{GL}_n(q)\) which have matrices from \(\mathrm{GL}_{\lambda_1}(q)\), \dots , \(\mathrm{GL}_{\lambda_r}(q)\) along the main diagonal, arbitrary elements above and zeros below that diagonal. In the reviewed paper, it is shown that \(P_{\lambda}\)-suitability is a strong enough tool to find all \(p\)-groups that are vertices of components of \(kP_{\lambda}\!\uparrow ^G\). All \(P_{\lambda}\)-suitable groups are found up to \(G\)-conjugation and it is shown that all \(P_{\lambda}\)-suitable groups are a vertex of exactly one component of \(kP_{\lambda}\!\uparrow ^G\).