Orthogonal units of the bifree double Burnside ring

الأصل · EN

The bifree double Burnside ring BΔ(G,G) of a finite group G has a natural anti-involution. We study the group BΔ∘(G,G) of orthogonal units in BΔ(G,G). It is shown that this group is always finite and contains a subgroup isomorphic to B(G)× (G), where B(G)× denotes the unit group of the Burnside ring of G and (G) denotes the outer automorphism group of G. Moreover it is shown that if G is nilpotent then BΔ∘(G,G) B(G)× (G). The results can be interpreted as positive answers to questions on equivalences of p-blocks of group algebras in the case that the block is the group algebra of a p-group.