Vol. 43, No. 3, 1972

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Classes of unimodular abelian group matrices

Dennis Garbanati and Robert Charles Thompson

Vol. 43 (1972), No. 3, 633–646

Let G be a finite abelian group, let G0 be the set of unimodular group matrices for G with rational integer entries, let G1 be the symmetric members of G0, and G2 the positive definite symmetric members of G0. Let K be either G1 or G2. On K impose the equivalence relation of group matrix congruence by asserting A B (for A,B K) if and only if C G0 exists such that A = CBC𝒯, where 𝒯 denotes transposition. M. Newman has estimated the number of classes under this equivalence relation, when G is cyclic. In this paper his study is continued for abelian groups. As part of the results it is shown that the class number of K is always a power of two, and when K is G1 the exact value of this class number is obtained. When K is G2 an upper bound for class number is found and shown to be sharp by exhibiting an infinite class of groups for which it is achieved.

Mathematical Subject Classification
Primary: 20E35
Received: 1 June 1971
Published: 1 December 1972
Dennis Garbanati
Robert Charles Thompson