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.
