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The group ring for S₃ Botta, Earle Peter

Abstract

The units in the group ring for S₃ over the integers are investigated. It is shown that the only units of finite order are of order two, three or six. Infinite classes of units of each of these orders are constructed as well as an infinite class of units of infinite order. The equation G = AA T, where G is a unimodular group matrix of rational integers and A a matrix of rational integers, is investigated in the ring of group matrices for S₃. It is shown that A = CP, where C is a unimodular group matrix of rational integers and P a generalized permutation matrix. It is also shown that if H is a positive definite symmetric unimodular group matrix then H = H₁H₁ T where H₁ is a group matrix of rational integers and H is of infinite order except in the trivial case when H = I.

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