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A Remark on the Units of Finite Order in The Group Ring of a Finite Group

Published online by Cambridge University Press:  20 November 2018

Gerald Losey
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Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will prove

Let G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.

In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 129 - 130
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Berman, S. D., On the equation xm = l in an integral group ring, Ukrain. Math. Z., 7 (1955), pp. 253-261.Google Scholar
2. Cohn, J. A. and Livingstone, D., On the structure of group algebras, I, Canadian J. Math., 17 (1965), pp. 583-593.Google Scholar
3. Dietzmann, A. P., Uberp-gruppen, Doklady Akad. Nauk SSSR, 15 (1937), pp. 71-76.Google Scholar