Minimal Product Sets in Finite Metacyclic Groups

Source abstract

Minimum product-set problems ask how small a product of two subsets of prescribed cardinalities can be in a finite group. Although the problem is completely understood for finite abelian groups, exact results in the nonabelian setting are available only for particular classes. We determine the minimum product-set function $μ_G$ for every finite metacyclic group, without assuming that the relevant cyclic extension splits. Our proof converts the extremal problem into a finite recursive arithmetic description of the feasible cardinality vectors of subset trios. The recursion combines the subgroup structure of finite metacyclic groups with the structural classification of maximal critical subset trios. The resulting characterization recovers the previously known formulas for abelian and nilpotent groups, dihedral groups, and nonabelian groups of order $pq$. It also yields metacyclic examples in which the minimum product-set size lies strictly between the classical lower and upper arithmetic bounds for finite solvable groups. For faithful split metacyclic groups, we further show that requiring one factor to be symmetric, contain the identity, and generate the whole group can force a strictly larger product set than the unrestricted minimum. The latter constraint admits a vertex-isoperimetric interpretation on connected Cayley graphs.