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Torsion units in integral group rings.

1991; De Gruyter; Volume: 1991; Issue: 415 Linguagem: Inglês

10.1515/crll.1991.415.175

1435-5345

Geometric and Algebraic Topology

Special cases of Bovdi's conjecture are proved.In particular the conjecture is proved for supersolvable and Frobenius groups.We also prove that if exp(G/Z>) is finite, a e VZG a torsion unit and m the smallest positive integer such that a" 1 G G then m divides exp(G/ Z).Let G be a group and let VZG be the group of units of augmentation one of the integral group ring ZG.Given an element x = J2x(g)g G ÏG we set= l(mod/?)and T^\x) = 0(modp) for j < n.In particular there is an element g G G such that o{x) = o(g).Considering these statements he conjectured that if JC is as in Lemma 1 then BC1: T^\x) = 1 and T^\x) = 0 for/ < n.In [4] BC1 is proved for metabelian nilpotent groups and in [2] it is proved in general for nilpotent groups.Bovdi also conjectured the following [1]: BC2: Letw = exp(G/Z(G)) be finite, where Z(G) denotes the center of G.If a G VZG is a torsion unit and m is the smallest positive integer such that a™ e G, then m divides n.We recall that H. J. Zassenhaus had conjectured the following: ZC1: Let G be a finite group and a G VZG a torsion unit then a is conjugated in QG, to an element of G. Lemma 1.1 below shows that ZC1 implies BC1.In this paper we deal with the conjectures BC1 and BC2 and show that BC1 holds for Frobenius groups and polycyclic groups whose commutator subgroup is nilpotent.In particular we re-obtain the result of [2] that BC1 holds for nilpotent groups.Also, we show that BC2 is true for all groups.In the text we denote by S nj -the Kronecker delta function which is 0 if/ ^ n and 1 if j = n.