Guilherme Leal

CENTRAL IDEMPOTENTS IN THE RATIONAL GROUP ALGEBRA OF A FINITE NILPOTENT GROUP

We describe the primitive central idempotents of a rational group algebra of a finite nilpotent group. The description does not make use of the character table of the group G.

Units of integral group rings of hamiltonian groups

We give a finite set of generators for a subgroup of finite index of the unit group of an integral group ring of an Hamiltonian group the order of which is only divisible by 2 and primes that are congruent to 3 modulo 8. An application is given to the unit group of the integral group ring of nilpotent groups.

Associative subalgebras of the octonians

In the mid 1970s, Michel Racine classified the maximal subalgebras of an octonian algebra. In this paper, we classify the maximal associative subalgebras. It turns out that there are four, up to isomorphism, all of dimension 4. In final sections, we apply our findings to investigate the groups that sit inside the Moufang loop of invertible elements of the split octonians and also to show that a well-known inequality of Jorgensen holds in a new context.

Central Units, Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S n in characteristic zero, and we show that they are units in very special instances.

Congruence subgroups in group rings

For a large class of groups G a precise congruence subgroup of the group generated by the bicyclic units of the integral group ring ZG is determined. As an application an upper bound is calculated for the index in the unit group of ZG for the group generated by the Bass cyclic units and the bicyclic units.