A note on the unit group of 𝑍𝑆₃

Allen, P. J.; Hobby, C.

doi:10.1090/S0002-9939-1987-0866420-X

A note on the unit group of $\textbf {Z}S_ 3$
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by P. J. Allen and C. Hobby PDF

Proc. Amer. Math. Soc. 99 (1987), 9-14 Request permission

Abstract:

The group $V(Z{S_3})$ of units of augmentation 1 in $Z{S_3}$ is characterized as the group of all doubly stochastic matrices in ${\text {GL(}}3,Z)$. Two normal complements are presented, one of which is torsion free while the other contains elements of order 2.

References

P. J. Allen and C. Hobby, A characterization of units in $\textbf {Z}[A_{4}]$, J. Algebra 66 (1980), no. 2, 534–543. MR 593609, DOI 10.1016/0021-8693(80)90102-7
G. H. Cliff, S. K. Sehgal, and A. R. Weiss, Units of integral group rings of metabelian groups, J. Algebra 73 (1981), no. 1, 167–185. MR 641639, DOI 10.1016/0021-8693(81)90353-7
I. Hughes and K. R. Pearson, The group of units of the integral group ring $ZS_{3}$, Canad. Math. Bull. 15 (1972), 529–534. MR 325743, DOI 10.4153/CMB-1972-093-1

The units of the integral group ring

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Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
D. S. Passman and P. F. Smith, Units in integral group rings, J. Algebra 69 (1981), no. 1, 213–239. MR 613869, DOI 10.1016/0021-8693(81)90139-3
Olga Taussky, Matrices of rational integers, Bull. Amer. Math. Soc. 66 (1960), 327–345. MR 120237, DOI 10.1090/S0002-9904-1960-10439-9