Torsion units of integral group rings of metacyclic groups

Luthar, I. S. ; Bhandari, A. K. (1983) Torsion units of integral group rings of metacyclic groups Journal of Number Theory, 17 (2). pp. 270-283. ISSN 0022-314X

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Let VZ G (respectively, VQ G) denote the group of units of augmentation 1 in the integral (respectively, rational) group ring of a finite group G. It has been conjectured [[7.]] that each element of finite order of VZ G is conjugate in VQ G to an element of G (see also [1.] and [6.]). To the best of our knowledge, the only nonabelian case (other than the Hamiltonian 2-groups) where this conjecture has been verified is G = S3 [[5.], 529-534]. In this paper this conjecture is verified for the metacyclic group G = < σ , τ: σp = 1 = τq, τστ-1 = σj> (p, q primes, p = 1 mod q, jq = 1, j N= 1 mod p) by expressing VZ G and VQ G as semidirect products of groups of q × q matrices. Although S. Galovitch, I. Reiner, and S. Ullom [Mathematika. 19 (1972), 105-111] obtained a description of VZ G, the discussion of torsion units was not attempted by them.

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