Torsion Units in Integral Group Rings of Certain Metabelian Groups

Abstract

It is shown that any torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to an element of ±G if G = XA with A a cyclic normal subgroup of G and X an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

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].…”

Torsion Units in Integral Group Ring of the Mathieu Simple GroupM22

“…The same method has also proved to be useful for some groups containing non-trivial normal subgroups. For some recent results we refer to [5,7,14,16,15,17]. Some related properties and weakened variations of the Zassenhaus conjecture can be found in [1,3,19].…”

Integral group ring of Rudvalis simple group

“…Since the group G possesses elements of orders 2, 3, 4, 5, 6,7,8,10,12,13,14,15,16,20,24,26, and 29, first of all we investigate units of some of these orders (except the units of orders 4, 6, 8, 10, 12, 14, 15, 16, 20, 24, and 26 ). After this, by Proposition 4, the order of each torsion unit divides the exponent of G, so to prove the Kimmerle conjecture it remains to consider units of orders 21, 35, 39, 58, 65, 87, 91, 145, 203, and 377.…”