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We construct a differential graded algebra (DGA) modelling certain $A_\infty$algebras associated with a finite group $G$ with cyclic Sylow subgroups, namely$H^*BG$ and $H_*\Omega BG^{^\wedge}_p$. We use our construction to investigatethe singularity and cosingularity categories of these algebras. We give acomplete classification of the indecomposables in these categories, anddescribe the Auslander--Reiten quiver. The theory applies to Brauer treealgebras in arbitrary characteristic, and we end with an example incharacteristic zero coming from the Hecke algebras of symmetric groups.

The singularity and cosingularity categories of $C^*BG$ for groups with cyclic Sylow $p$-subgroups