We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring ZG of a finite nilpotent group G has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators.

We compute the rank of the group of central units in the integral group ring ZG 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 ZG 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 QG.Finally, we apply these results to describe finitely many generators of a subgroup of finite index in the group of units of ZG, this for metacyclic groups G of the form G = C q m ⋊ 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 .

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Abstract. Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra FG for a large class of groups G.As an illustration of our methods, alternative constructions to some best linear codes over F 2 and F 3 are given. Furthermore, we give constructions of non-abelian left group codes.

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.