Orthogonal units of the bifree double Burnside ring

TL;DR

This paper investigates the structure of orthogonal units in the bifree double Burnside ring of a finite group, revealing their finiteness and relation to the Burnside ring's units and automorphisms, especially for nilpotent groups.

Contribution

It characterizes the group of orthogonal units in the bifree double Burnside ring, showing it is finite and explicitly describes its structure for nilpotent groups.

Findings

The group of orthogonal units is always finite.

Contains a subgroup isomorphic to the semi-direct product of the Burnside ring units and outer automorphisms.

For nilpotent groups, the orthogonal units group is isomorphic to this semi-direct product.

Abstract

The bifree double Burnside ring $B^\Delta(G,G)$ of a finite group $G$ has a natural anti-involution. We study the group $B^\Delta_\circ(G,G)$ of orthogonal units in $B^\Delta(G,G)$. 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 $B^\Delta_\circ(G,G)\cong B(G)^\times\rtimes \Out(G)$. 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.