Cyclic cellularity and active sums

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.