Group algebras of Kleinian type and groups of units

The algebras of Kleinian type are finite dimensional semisimple rational algebras $A$ such that the group of units of an order in $A$ is commensurable with a direct product of Kleinian groups. We classify the Schur algebras of Kleinian type and the group algebras of Kleinian type. As an application, we characterize the group rings $RG$, with $R$ an order in a number field and $G$ a finite group, such that $RG^*$ is virtually a direct product of free-by-free groups.Comment: 12 page

On idempotents and the number of simple components of semisimple group algebra

We describe the primitive central idempotents of the group algebra over a number field of finite monomial groups. We give also a description of the Wedderburn decomposition of the group algebra over a number field for finite strongly monomial groups. Further, for this class of group algebras, we describe when the number of simple components agrees with the number of simple components of the rational group algebra. Finally, we give a formula for the rank of the central units of the group ring over the ring of integers of a number field for a strongly monomial group

Group rings of finite strongly monomial groups: central units and primitive idempotents

We compute the rank of the group of central units in the integral group ring $\Z G$ of a finite strongly monomial group $G$. The formula obtained is in terms of the strong Shoda pairs of $G$. Next we construct a virtual basis of the group of central units of $\Z G$ for a class of groups $G$ properly contained in the finite strongly monomial groups. Furthermore, for another class of groups $G$ inside the finite strongly monomial groups, we give an explicit construction of a complete set of orthogonal primitive idempotents of \Q G. Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of $\Z G$, this for metacyclic groups $G$ of the form $G=C_{q^m}\rtimes C_{p^n}$ with $p$ and $q$ different primes and the cyclic group $C_{p^n}$ of order $p^n$ acting faithfully on the cyclic group $C_{q^m}$ of order $q^m$

On the Congruence Subgroup Problem for integral group rings

Let $G$ be a finite group, $\Z G$ the integral group ring of $G$ and \U(\Z G) the group of units of $\Z G$. The Congruence Subgroup Problem for \U(\Z G) is the problem of deciding if every subgroup of finite index of \U(\Z G) contains a congruence subgroup, i.e. the kernel of the natural homomorphism \U(\Z G) \rightarrow \U(\Z G/m\Z G) for some positive integer $m$. The congruence kernel of \U(\Z G) is the kernel of the natural map from the completion of \U(\Z G) with respect to the profinite topology to the completion with respect to the topology defined by the congruence subgroups. The Congruence Subgroup Problem has a positive solution if and only if the congruence kernel is trivial. We obtain an approximation to the problem of classifying the finite groups for which the congruence kernel of \U(\Z G) is finite. More precisely, we obtain a list $L$ formed by three families of finite groups and 19 additional groups such that if the congruence kernel of \U(\Z G) is infinite then $G$ has an epimorphic image isomorphic to one of the groups of $L$. About the converse of this statement we at least know that if one of the 19 additional groups in $L$ is isomorphic to an epimorphic image of $G$ then the congruence kernel of \U(\Z G) is infinite. However, to decide for the finiteness of the congruence kernel in case $G$ has an epimorphic image isomorphic to one of the groups in the three families of $L$ one needs to know if the congruence kernel of the group of units of an order in some specific division algebras is finite and this seems a difficult problem.Comment: 28 page

COMPUTING THE WEDDERBURN DECOMPOSITION OF GROUP ALGEBRAS BY THE BRAUER–WITT THEOREM

Abstract. We present an alternative constructive proof of the Brauer–Witt theorem using the so-called strongly monomial characters that gives rise to an algorithm for computing the Wedderburn decomposition of semisimple group algebras of finite groups. 1