Homology of certain sets of 2-subgroups of symmetric groups
In his paper [Adv. Math. 28, 101-128 (1978; Zbl 0388.55007)], \textit{D. Quillen} showed that the set of non-trivial \(p\)-subgroups of a finite group, \(s_ p(G)\), has the same homotopy type as the set of non-trivial elementary abelian \(p\)-subgroups. In this paper the author initiates a study of the group \(d_ 2(n)\) of non-trivial elmentary abelian 2- subgroups of the symmetric group \(S_ n\), and relates it to \(s_ 2(S_ n)\). Much of the remainder of the paper is devoted to homology calculations. For example it is found that certain of these groups are cyclic of order 3. The article concludes with the observation that although it is difficult to compute the homology of \(d_ 2(n)\) exactly, it is easy to find its Lefschetz module.
scientific article; zbMATH DE number 63899