The combinatorics and the homology of the poset of subgroups of \(p\)-power index
In this quite multidimensional paper, the objects of interest are posets \(S^ p(G)\) of all subgroups \(H \neq G\) (a finite group) of prime power index (or generalizations to \(S^ \pi (G)\), \(\pi\) a set of primes), studied mainly through considerations of the order complex \(CS^ p(G)\) of \(S^ p(G)\) and its geometric realization \(| CS^ p(G)|\), which is shown to be a homotopy wedge of spheres of a fixed (interesting) dimension. Hence also the Möbius number \(\mu(S^ p(G))\) is obtained, thereby providing means to represent \(G\) on \(CS^ p(G)\) which in turn permits a representation of \(G\) realized as a set of matrices with entries in \(\{-1, 0, +1\}\). The authors draw on a wealth of results from the theory of finite groups along with the algebraic-combinatorial theory involved in the shellability theory of posets and their complexes as well as their own earlier results of a related nature, some dealing with subgroup posets different from those discussed here. Significant by-products include a proof of a dual of a conjecture of Quillen (i.e., \(S^ p(G)\) is \(G\)-contractible iff contractible iff \(\mu(S^ p(G)) = 0\) iff \(O^ p(G)\), the minimal normal subgroup of \(p\)- power index, \(\neq G\)) and a conjecture that \(S^ p_ c(G)\), a closely associated subposet of \(S^ p(G)\), \(G\)-homotopy equivalent to \(S^ p(G)\), is CL-shellable for all finite groups \(G\), based on observations made in arguments in the text of the paper. From the proofs and statements of the results, it is clear that ``comparability'' is possibly as important a property as actual ``order'' once the group \(G\) and the prime \(p\) have been fixed. En fin the paper makes a contribution both to the theory of finite groups and to the topological combinatorics of posets.
scientific article; zbMATH DE number 500586