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 ${\sup }_{\omega (G)\le n}a(G)=2^{\lfloor (n-1)/2\rfloor }+1$ for every $n\ge 63$. 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.

arXiv comment21 pages. Comments welcome