# Orthogonal units of the double Burnside ring

## TL;DR

This paper investigates the structure of orthogonal units within the double Burnside ring of a finite group, providing new insights into their properties and explicit descriptions for cyclic p-groups.

## Contribution

It introduces an inflation map for orthogonal units and characterizes these units for cyclic p-groups, advancing understanding of the double Burnside ring's unit subgroup.

## Key findings

- Defined and studied orthogonal units in the double Burnside ring.
- Established an inflation map linking units of quotient groups.
- Determined orthogonal units for cyclic p-groups with odd primes.

## Abstract

Given a finite group $G$, its double Burnside ring $B(G,G)$, has a natural duality operation that arises from considering opposite $(G,G)$-bisets. In this article, we systematically study the subgroup of units of $B(G,G)$, where elements are inverse to their dual, so called orthogonal units. We show the existence of an inflation map that embeds the group of orthogonal units of $B(G/N,G/N)$ into the group of orthogonal units of $B(G,G)$, when $N$ is a normal subgroup of $G$, and study some properties and consequences. In particular, we use these maps to determine the orthogonal units of $B(G,G)$, when $G$ is a cyclic $p$-group, and $p$ is an odd prime.