Eventual Exactness and Rigidity for Abelian Covers of Groups

Abstract

For a group $G$, let $a(G)$ denote the minimum number of abelian subgroups covering $G$, and let $\omega(G)$ be the maximum size of a pairwise noncommuting set. We prove the exact formula $\sup_{\omega(G)\leq n}a(G)=2^{\lfloor(n-1)/2\rfloor}+1$ for every $n\geq 63$. Equality holds precisely for groups with derived subgroup of order two and binary symplectic central quotient of dimension $2\lfloor(n-1)/2\rfloor$. The proof combines centralizer descent with a structural separation of nonscalar alternating commutator maps.