Combinatorial properties of intervals in finite solvable groups
Given a finite solvable group \(G\) and a subgroup \(H \leq G\), the interval \([H,G] = \{U \mid H \leq U \leq G\}\) in the subgroup poset is the object of interest in this quite penetrating paper in which the author has successfully introduced important combinatorial principles which combined with suitable group-theoretical results provide quite powerful means of analysis leading to a substantial collection of new results as well as to alternative proofs of several results in the area which are well-known there. The approach used depends on the fact that if \(\mu[H,G]\) denotes the Möbius number of the interval and if \(k(G)\) denotes the poset of all subgroups \(H\) of \(G\) such that \(\mu[H,G] = 0\), then \(\mu[H,G] = \mu[H,G]_k\), when \([H,G]_k = (k(G) \cap [H,G]) \cup \{G\}\). Furthermore, if \(\mu[H,G] = 0\), then a considerable effort is sufficient to show that \([H,G]_k\) is a lexicographically shellable poset whose shelling is determined and then permits determination of the Möbius number in terms of the numbers of maximal subgroups in the interval. Hence, e.g., a proof plus further insight into a vintage result of Gaschütz is provided. From \([H,G]\) and \([H,G]_k\) passage is made to \([H,G]/G\) and \([H,G]_k/G\) whose elements consist of conjugacy classes \([U] = \{U^g\mid g \in G\}\) of subgroups with \([U] \leq [V]\) iff \(U^g \leq V\) for some \(g \in G\). The approach already used is employed to determine the Möbius number \(\mu([H,G]_k / G)\) in terms of conjugacy classes of maximal subgroups, thence passing to \(\mu([H,G]/G)\) by equality. Knowledge of the length of orbits in this setting produces the important formula \(|G : N_G(H) |\mu[H,G] = |HG : H|\mu([H,G]/G)\) which is analyzed and refined further to obtain several consequences of interest also.
scientific article; zbMATH DE number 757790
scientific article