Tuesday, January 29, 15:15, 7.527
Martin Hertweck, Rational and p-adic conjugacy of p-torsion units
in finite integral group rings.
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
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.)