Papers › Abelianization of the unit group of an integral group ring
Abelianization of the unit group of an integral group ring
Andreas Bächle, Sugandha Maheshwary, Leo Margolis
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For a finite group G and U: = U(ℤG), the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization U/U′ of U. We pose questions on the connections between the exponent of G/G′ and the exponent of U/U′ as well as between the ranks of the torsion-free parts of Z(U), the center of U, and U/U′. We show that the units originating from known generic constructions of units in ℤG are well-behaved under the projection from U to U/U′ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ^+18].
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