On the Torsion Units of Some Integral Group Rings

Abstract

Abstract. It is shown that any torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a trivial unit if G = P A with P a normal Sylow p-subgroup of G and A an abelian p -group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture holds for PSL(2, 7), the finite simple group of order 168.

Integral Group Ring of the Mathieu Simple GroupM23

“…Also some related properties and some weakened variations of the Zassenhaus conjecture as well can be found in [1,22] and [3,20]. For some recent results we refer to [5,7,15,16,17,18].…”

“…(see [15], Proposition 3.1; [16], Proposition 2.2) Let G be a finite group and let u be a torsion unit in V (ZG). If x is an element of G whose p-part, for some prime p, has order strictly greater than the order of the p-part of u, then ε x (u) = 0.…”

Finite groups of units and their composition factors in the integral group rings of the groups PSL(2, q)

“…Note that xð1Þ ¼ q þ e. The group G has only one conjugacy class of involutions. Let s be an involution in G. By [14,Corollary 3.5], it follows that an involution in H is conjugate to s by a unit in QG. So xðxÞ ¼ xðsÞ for an involution x of H. Note that xðsÞ ¼ À2e.…”

“…For known results on this still unsolved conjecture the reader is referred to [23,Chapter 5], [24, §8] and [14], [15]. In fact this conjugacy question makes sense even for finite groups of units in ZG.…”

Torsion units for a Ree group, Tits group and a Steinberg triality group

“…We combine the Luthar-Passi Method together with techniques developed in [14,37] to obtain our results. Our main results are as follows:…”