  Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} =
\displaystyle\prod^n_{i=1} \Gamma_i$ and $b>1,$ we define a polymatroid
$P(\mathcal{H},b).$ If all of the $\Gamma_i$ are isomorphic to
$\mathbb{Z}/p\mathbb{Z},$ $p$ a prime, and $b=p,$ then $P(\mathcal{H},b)$ is
the usual matroid associated to any $\mathbb{Z}/p\mathbb{Z}$-matrix whose row
space equals $\mathcal{H}.$ In general, there are many ways in which the
relationship between $P(\mathcal{H},b)$ and $\mathcal{H}$ 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 $P(\mathcal{H},b),$ analogs of Greene's theorem and the MacWilliams
identities when $\mathcal{H}$ is a group code over a nonabelian group, and a
connection to the combinatorial Laplacian of a quotient space determined by
$\mathcal{G}$ and $\mathcal{H}.$ 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.
