Polymatroids are to finite groups as matroids are to finite fields

Given a subgroup H of a product of finite groups G = ∏ⁿᵢ₌₁ Γᵢ and $b>1,$ we define a polymatroid P(H,b). If all of the Γᵢ are isomorphic to Z/pZ, p a prime, and $b=p,$ then P(H,b) is the usual matroid associated to any Z/pZ-matrix whose row space equals H. In general, there are many ways in which the relationship between P(H,b) and H mirrors that of the relationship between a matroid and a subspace of a finite vector space. These include representability by excluded minors, the Crapo-Rota critical theorem, the existence of a concrete algebraic object representing the polymatroid dual of P(H,b), analogs of Greene's theorem and the MacWilliams identities when H is a group code over a nonabelian group, and a connection to the combinatorial Laplacian of a quotient space determined by G and H. We use the group Crapo-Rota critical theorem to demonstrate an extension to hypergraphs of the classical duality between proper colorings and nowhere-zero flows on graphs.