Counting conjugacy classes of cyclic subgroups for fusion systems

TL;DR

This paper provides a new proof and a fusion system version of a result relating conjugacy classes of cyclic subgroups to matrix ranks derived from characteristic bisets and idempotents.

Contribution

It introduces a fusion system analogue of Thévenaz's observation, linking conjugacy class counts to matrix ranks in the context of saturated fusion systems.

Findings

Number of conjugacy classes equals the rank of specific matrices.

Matrix ranks are derived from characteristic bisets and idempotents.

Results connect fusion system properties with algebraic matrix invariants.

Abstract

We give another proof of an observation of Th\'evenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of cyclic subgroups of $S$ is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system $\CF$ as in our previous work \cite{P2010}.