# Finite groups of rank two which do not involve $Qd(p)$

## TL;DR

This paper characterizes finite groups of rank two that do not involve the group $Qd(p)$, showing a specific equivalence related to $p'$-involvement and applying this to construct finite group actions on mod-$p$ homotopy spheres.

## Contribution

It establishes a criterion for when finite groups of rank two involve $Qd(p)$, refining understanding of their structure and applications to group actions on homotopy spheres.

## Key findings

- Galois involvement of $Qd(p)$ characterized for rank two groups.
- Extension of Glauberman's ZJ-theorem for these groups.
- Counterexample provided for primes $p \\leq 3$.

## Abstract

Let $p>3$ be a prime. We show that if $G$ is a finite group with $p$-rank equal to 2, then $G$ involves $Qd(p)$ if and only if $G$ $p'$-involves $Qd(p)$. This allows us to use a version of Glauberman's ZJ-theorem to give a more direct construction of finite group actions on mod-$p$ homotopy spheres. We give an example to illustrate that the above conclusion does not hold for $p \leq 3$.