Let FG denote the group algebra of a group G with coefficients on a field F. The group algebra FG codifies the representation theory of G over the field F and it is therefore a fundamental tool to understand essential properties of G.
By Masckhe Theorem, RG is semisimple if and only if G is finite of order not multiple of the characteristic of F.
In that case, RG is a direct product of simple algebras and these simple algebras are matrix algebras over division algebra, by Wedderburn Theorem.
Moreover these simple algebras are cyclotomic over F, that is they are crossed products of cyclotomic extensions of F such that the values taken by the cocycle are roots of unity. The expression of FG as a direct product of simple algebras is known as the Wedderburn decomposition of FG.
Knowing the Wedderburn decomposition is quite useful because it gives information on for example the group of automorphisms of FG or the group of units of some groups of units. In this talk we will explain an effective method to compute the Wedderburn of semisimple group algebras which has been implemented in a package for the computer system GAP. We will show examples of how to use the programs of the package.
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