Coset complexes of 𝑝-subgroups in finite groups

Let 𝐺 be a finite group and 𝑝 a prime. We denote by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Cₚ(G) the poset of all cosets of 𝑝-subgroups of 𝐺. We characterize the homotopy type of the geometric realization <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi mathvariant="normal">Δ</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> ₚ(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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> χ(Cₚ(G)) satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:msup> <m:mi>p</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> </m:mrow> </m:math> χ(Cₚ(G))= G_{p^{}} . Additionally, we demonstrate that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>χ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi mathvariant="script">C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≡</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:msup> <m:mi>p</m:mi> <m:mo>′</m:mo> </m:msup> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>mod</m:mi> <m:mo lspace="0.500em">⁢</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> χ(Cₚ(G))≡ G_{p^{}}\ (mod\ p) for any group 𝐺.