A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. Let A ∘ B A \circ B denote the Hadamard product of matrices A and B. The set of groups G of nonsingular n × n n \times n matrices which contain the diagonal group D \mathcal {D} and such that for every pair A, B of matrices in G we have A ∘ B T ∈ D A \circ {B^T} \in \mathcal {D} is denoted by A n {\mathcal {A}_n} . If the underlying field has at least three elements then A n {\mathcal {A}_n} consists of permanent groups. A partial converse is available: If a permanent group G is generated by D \mathcal {D} together with a set S of elementary matrices and a set Q of permutation matrices then G = H K G = HK where H is the subgroup generated by Q and K is generated by D \mathcal {D} and S, and K ∈ A n K \in {\mathcal {A}_n} .