Blocks of defect 1 and units in integral group rings
Over the decades that U(ZG), the unit group of the integral group ring of a finite group G, has been studied, many conjectures have been raised on how the structure of G influences the structure of subgroups of U(ZG). Though it often took considerable time, counterexamples for the strongest of these conjectures were found in the class of solvable groups. Contrary to this, the arithmetic properties of finite subgroups of U(ZG) are very restricted for solvable G. For instance, the orders of group elements and orders of torsion units u in U(ZG) coincide, under the natural assumption 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 distinct primes, that G must also 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 the combinatorics of Young tableaux 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 for 22 sporadic simple groups and also for infinite series of almost simple groups of Lie type.
This is joint work with M. Caicedo.