In joint work with Russ Woodroofe, we showed that the order complex of the poset of all cosets of all proper subgroups of a finite group, ordered by inclusion, has noncontractible order complex using Smith Theory. A key part of our proof involves invariable generation of finite groups: two subsets $S,T$ of a group $G$ generate $G$ invariably if, for every $g,h \in G$, $g^{-1}Sg$ and $h^{-1}Th$ together generate $G$. It remains open whether the alternating group $A_n$ can be generated invariably by $\{s\}$ and $\{t\}$ with both $s,t$ having prime power order. This question is closely related to a (still open) question about prime divisors of binomial coefficients. I will discuss all of this, along with current work joint with Bob Guralnick and Russ Woodroofe about invariable generation of arbitrary simple groups by two elements of prime or prime power order.