Stable homotopy classification of \(BG_ p^ \land\)
The authors present proofs of their results announced in an earlier note [Bull. Am. Math. Soc., New Ser. 27, No. 1, 165-170 (1992; Zbl 0754.55015)]. The main theorem is that, given a prime number \(p\), for two finite groups \(G\) and \(G'\) the \(p\)-completions of their classifying spaces \(BG\) and \(BG'\) (or, equivalently, the \(p\)- localizations of \(BG\) and \(BG'\) have the same stable homotopy type if and only if any one of the following two equivalent conditions holds: (1) For every \(p\)-group \(Q\), the \(\mathbb{F}_p\)-vector spaces \(\mathbb{F}_p \text{Rep} (Q,G)\) and \(\mathbb{F}_p \text{Rep} (Q,G')\) are isomorphic as \(\text{Out} (Q)\)-modules; here the basis \(\text{Rep} (Q,G)\) of \(\mathbb{F}_p \text{Rep} (Q,G)\) consists of \(G\)-conjugacy classes of homomorphisms from \(Q\) into \(G\). (2) For every \(p\)-group \(Q\), \(\mathbb{F}_p \text{Inj} (Q,G)\) and \(\mathbb{F}_p \text{Inj} (Q,G')\) are isomorphic as \(\text{Out} (Q)\)- modules; here \(\text{Inj} (Q,G) \subseteq \text{Rep} (Q, G)\) consists of conjugacy classes of injective homomorphisms. In the proof, the authors essentially use their results obtained in a previous paper [Topology 31, No. 1, 143-156 (1992; Zbl 0752.55010)]. The concept of linkage allows them to use representation theory in describing the stable summands of the \(p\)-localization of \(BG\); the multiplicity of a stable summand turns out to be the rank of a certain matrix. This matrix then allows to determine the stable homotopy type of the \(p\)- localization of \(BG\). Several interesting applications of the main theorem, primarily to the case when \(G\) and \(G'\) are finite groups with normal Sylow \(p\)- subgroups, are also proved in this paper.
finite groups with normal Sylow \(p\)-subgroups
scientific article; zbMATH DE number 794443