Martin Hertweck, Rational and p-adic conjugacy of p-torsion units in finite integral group rings.

Abstract:

This talk is about units of finite order (torsion units) in the group ring RG of a finite group G over an "integral" ring R (such as the rational integers Z).

A conjecture of Hans Zassenhaus from the 1970s asserts that every torsion unit in ZG is conjugate to an element of G or -G by a unit in the rational group ring QG.

The outstanding result in the field is Al Weiss's proof (from 1988, 1991) that for a nilpotent group G, the conjecture is true. Weiss even proved (for nilpotent G) that a finite subgroup H of V(ZG), the group of augmentation one units of ZG, is conjugate to a subgroup of G by a unit in QG, not without demonstrating that this, in his own words, "is still a rather crude description of the actual situation." We just mention that his results show that when the subgroup H is cyclic, conjugacy of H to a subgroup of G already takes place in the units of Z_pG, for each rational prime p, where Z_p denotes the p-adic integers ("p-adic conjugacy").

For p-torsion units in ZG, p-adic conjugacy, rather than rational conjugacy, is highly desirable, for example for better understanding of how they are embedded in ZG, or for inductive approaches to the Zassenhaus conjecture. (Unfortunately, p-adic conjugacy is not given in general, as easy examples show.)

To each torsion unit u in RG there is associated a bimodule G_RG_u which is RG, with G and < u > acting by multiplication from left and right, respectively. It seems that these bimodules are the right objects to deal with. We will take a closer look on the indecomposable summands of these bimodules.

As a consequence, we obtain, for arbitrary G, a criterion for a p-torsion unit in ZG to be p-adically conjugate to an element of G (which presupposes rational conjugacy), in terms of its associated bimodule. More precisely: When R is a p-adic ring with quotient field K, and u is a p-torsion unit in RG which is conjugate to an element x of G by a unit in KG, then u is conjugate to x by a unit in RG if and only if the bimodule G_RG_u is a direct summand of a direct sum of copies of G_RG_x.

This criterion applies to torsion units in ZG which map to the identity under the natural ring homomorphism ZG -> ZG/N for a normal p-subgroup N of G, to give p-adic conjugacy to elements of G. (Through application of the p-permutation module theorem of Al Weiss.)