## 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.

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