Orthogonal units of the bifree double Burnside ring

The bifree double Burnside ring $B^Δ(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B^Δ_\circ(G,G)$ of orthogonal units in $B^Δ(G,G)$. It is shown that this group is always finite and contains a subgroup isomorphic to $B(G)^\times\rtimes \Out(G)$, where $B(G)^\times$ denotes the unit group of the Burnside ring of $G$ and $\Out(G)$ denotes the outer automorphism group of $G$. Moreover it is shown that if $G$ is nilpotent then $B^Δ_\circ(G,G)\cong B(G)^\times\rtimes \Out(G)$. 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.

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