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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Permanent groups

Authors: Leroy B. Beasley and Larry Cummings
Journal: Proc. Amer. Math. Soc. 34 (1972), 351-355
MSC: Primary 15A15
MathSciNet review: 0419474
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Abstract: A permanent group is a group of nonsingular matrices on which the permanent function is multiplicative. Let $ A \circ B$ denote the Hadamard product of matrices A and B. The set of groups G of nonsingular $ n \times n$ matrices which contain the diagonal group $ \mathcal{D}$ and such that for every pair A, B of matrices in G we have $ A \circ {B^T} \in \mathcal{D}$ is denoted by $ {\mathcal{A}_n}$.

If the underlying field has at least three elements then $ {\mathcal{A}_n}$ consists of permanent groups. A partial converse is available:

If a permanent group G is generated by $ \mathcal{D}$ together with a set S of elementary matrices and a set Q of permutation matrices then $ G = HK$ where H is the subgroup generated by Q and K is generated by $ \mathcal{D}$ and S, and $ K \in {\mathcal{A}_n}$.

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Keywords: Permanent, matrix groups
Article copyright: © Copyright 1972 American Mathematical Society

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