Cyclic cellularity and active sums

TL;DR

The paper investigates conditions under which the active sum of a family of subgroups inherits cellularity properties related to cyclic groups, with applications to Coxeter groups and special linear groups.

Contribution

It establishes that if all subgroups are $C_n$-cellular with finite exponent dividing $n$, then their active sum is also $C_n$-cellular, providing new proofs for Coxeter and $ ext{SL}(n,q)$ groups.

Findings

Active sums inherit cyclic cellularity under certain conditions.

Coxeter groups are $C_2$-cellular.

Many $ ext{SL}(n,q)$ groups are $C_3$-cellular.

Abstract

Let $G$ be a group and let $\mathcal{F}$ be a family of subgroups of $G$ closed under conjugation. For a positive integer $n$, let $C_n$ denote a cyclic group of order $n$. We show that if there exists an integer $n$ such that every group in $\mathcal{F}$ is $C_n$-cellular and has finite exponent diving $n$, then the active sum $S$ of $\mathcal{F}$ is $C_n$-cellular. We obtain a couple of interesting consequences of this result, using results about cellularity. Finally, we give different proofs of the facts that Coxeter groups are $C_2$-cellular and that many groups of the form $\mathrm{SL}(n,\,q)$ for $n\geq3$ are $C_3$-cellular.