Shoda pairs of finite groups

Abstract

A fundamental problem in group algebras is to determine the complete set of primitive central idempotents and precise description of Wedderburn decomposition of semisimple group algebras. The problem is of great interest due to its relations to various other problems in group rings. The classical approach is computationally inefficient and does not allow an immediate description of Wedderburn decomposition of semisimple group algebras. A new direction to this investigation was provided by Olivieri, del Rio and Simon, in 2004, when they introduced the notion of Shoda pairs. A significant result proved by them is the explicit description of the Wedderburn decomposition of QG, when G is strongly monomial. This research shifted the focus of group ring theorists to understand strongly monomial groups and to seek applications in understanding the unit group of integral group rings. In this thesis, generalized strongly monomial groups are defined which is a natural but non trivial generalization of strongly monomial groups. We provide an extensive list of well known classes of groups which are generalized strongly monomial; thus emphasizing that the class of generalized strongly monomial groups is an enormous class of monomial groups. We also study the class C of monomial groups consisting of all finite groups G in which each subquotient of G is either abelian or has a non central abelian normal subgroup and proved that the groups in class C are generalized strongly monomial. We provide a character free description of simple components of QG when G is generalized strongly monomial. Also, we provide a complete set of Shoda pairs using character triples for the groups in class C. Furthermore the work by Jespers, Olteanu, del Rio and Van Geldar, in 2014, on the construction of a subgroup of finite index in the group of central units of integral group ring of a strongly monomial group has been generalized. We also shifted our gears to understand the complete algebraic structure of semisimple finite group algebras.