A note on coset complexes of $p$-subgroups

This paper investigates the coset complexes of $p$-subgroups in finite groups. Given a finite group $G$ and a prime $p$, we define \( \mathscr{C}_p(G) \) as the poset of all cosets of $p$-subgroups of $G$. We construct a probability function \( P_p(G,s) \) with group-theoretic connections, strengthen the congruence formula of the $p$-local Euler characteristic of \( \mathscr{C}_p(G) \), and analyze the connectivity of \( \mathscr{C}_p(G) \).