On the cohomology and homotopy of Swan groups
If \(P\) is a Sylow \(p\)-subgroup of a finite group \(G\), then there is an injective map \(H^*(G;\mathbb{F}_p) @>\text{res}>> H^*(P;\mathbb{F}_p)^{W_G(P)}\), where \(W_G(P)=N_G(P)/PC_G(P)\) and cohomology is taken with simple coefficients in \(\mathbb{F}_p\). A Swan group is a finite \(p\)-group \(P\) for which this map is always an isomorphism, for any finite group \(G\) with \(P\) as Sylow \(p\)-subgroup. The authors establish a sufficient condition for a \(p\)-group to be a Swan group and show that for \(p\) odd, every metacyclic \(p\)-group is a Swan group. They then show that if the classifying space \(BP\) has no stable summands involving the transfer, then \(P\) is a Swan group. Finally, using U. Martin's definition of ``almost all'', they show that almost all \(p\)-groups are Swan groups.
scientific article; zbMATH DE number 1033934