Referent: Leo Margolis, Blocks of defect 1 and units in integral group rings

Donnerstag,  11.07.2019, 17:00 h

Ort:Raum 418-436

Over the decades that U(ZG), the unit group of an integral
group ring of a nite group G, has been studied, many conjectures have been
raised on how the structure of G inuences the structure of subgroups of G.
Though it took often considerable time, the strongest of these conjectures
found counterexamples in the class of solvable groups. Contrary to this
the arithmetic properties of nite subgroups of U(ZG) are very restricted
for solvable G. For instance the orders of group elements and orders of
torsion units u € U(ZG) coincide, under the natural assumptions that u has
augmentation 1.
A problem on these arithmetic properties, the Prime Graph Question for
integral group rings, asks whether it is true that whenever U(ZG) contains an
element of augmentation 1 and order pq, where p and q are dierent primes,
also G must contain an element of order pq. In contrast to other problems in
the area, this question is known to have a reduction to almost simple groups.
Employing Young tableaux combinatorics and Brauer's theory of blocks
of defect 1 we show that when the Sylow p-subgroup of G has order p then
U(ZG) contains an element of augmentation 1 and order pq, for any prime
q, if and only if G contains an element of order pq. This directly answers the
Prime Graph Question in particular for 22 sporadic simple groups and also
for innite series of almost simple groups of Lie type.
This is joint work with M. Caicedo.