What question did this study set out to answer?

To characterize the homotopy type of coset complexes associated with p-subgroups in finite groups and establish properties of Euler characteristics.

March 2, 2026

Coset complexes of 𝑝-subgroups in finite groups

Key Points

To characterize the homotopy type of coset complexes associated with p-subgroups in finite groups and establish properties of Euler characteristics.
Characterization of the poset of cosets of p-subgroups in a finite group G.
Analysis of the homotopy type and geometric realization of these cosets.
Investigation of Euler characteristics for specific classes of finite groups.
The Euler characteristic of coset complexes for p-closed groups yields a specific formula.
It was found that χ(Cp(G)) equals |G|_{p'} for certain finite groups.
The congruence χ(Cp(G)) ≡ |G|_{p'} (mod p) holds for all groups G.

Abstract

Abstract Let 𝐺 be a finite group and 𝑝 a prime. We denote by C p ⁢ (G) C₏ (G) the poset of all cosets of 𝑝-subgroups of 𝐺. We characterize the homotopy type of the geometric realization | Δ ⁢ C p ⁢ (G) | ₏ (G) for 𝑝-closed groups 𝐺, which is motivated by a question of K. S. Brown, and we characterize a class of finite groups for which the Euler characteristic χ ⁢ (C p ⁢ (G) ) (C₏ (G) ) satisfies χ ⁢ (C p ⁢ (G) ) = | G | p ′ (C₏ (G) ) = G₏^. Additionally, we demonstrate that χ ⁢ (C p ⁢ (G) ) ≡ | G | p ′ ⁢ (mod ⁢ p) (C₏ (G) ) G₏^\ (mod\ p) for any group 𝐺.

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Coset complexes of 𝑝-subgroups in finite groups

Key Points

Abstract

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