Derivations of group algebras with applications /

Abstract

This thesis is a study of derivations of group algebras. Derivations are shown to be trivial for semisimple group algebras of abelian groups. The derivations of a group algebra are classified in terms of the generators and defining relations of the group. If RG is a group ring, where R is commutative and S is a set of generators of G then necessary and sufficient conditions on a map from S to RG are established, such that the map can be extended to an R-derivation of RG. This theorem is utilised to construct a basis for the vector space of derivations of abelian group algebras, dihedral group algebras and quaternion group algebras. Derivations of group algebras are considered as linear finite dynamical systems and their associated directed graphs are studied. The motivation for this comes from the fact that if DerpKGq and DerpKHq are not isomor phic as additive groups then KG and KH are not isomorphic as rings. It is shown that if R and S are ring isomorphic, then there is a bijection from DerpRq onto DerpSq such that corresponding derivations have isomorphic associated digraphs. Therefore properties of the linear finite dynamical sys tem associated with a derivation can be used to distinguish between group rings. Derivations of a group algebra form a Lie algebra and it is shown that this Lie algebra DerpKGq is a complete Lie algebra, when G is a finite abelian group such that its Sylow p-subgroup is elementary abelian. Derivations can be used to show that two group algebras are not iso morphic as rings. As an example dihedral and quaternion group algebras are contrasted by showing that their respective derivation Lie algebras have different dimension and centers of different dimension. The thesis concludes by giving an alternative proof of Deskins’ Theorem using derivations.