Transitive fusion systems over a class of finite p-groups

TL;DR

This paper proves that a conjecture about transitive fusion systems over certain finite p-groups holds true when the p-rank of the group is 2, showing that such groups are either extraspecial of order p^3 or elementary abelian.

Contribution

The paper provides a simple proof confirming the conjecture for p-groups with p-rank 2, narrowing down the possible structures of such groups in transitive fusion systems.

Findings

Conjecture holds for p-rank 2 groups

Groups are either extraspecial of order p^3 or elementary abelian

Method simplifies the proof process

Abstract

Let $p$ be an odd prime and $S$ a nonabelian finite $p$-group. In [9, 10], they proposed the following conjecture: if $\mathcal{F}$ be a transitive fusion system over a finite $p$-group $S$, then $S$ is either extraspecial of order $p^{3}$ or elementary abelian. In this note, we use an easy method to prove that this conjecture holds when the $p$-rank of $S$ is 2.