Group Algebra Over A Finite Group
Group algebras occur naturally in the theory of group representations of finite groups. The group algebra K over a field K is essentially the group ring, with the field K taking the place of the ring. As a set and vector space, it is the free vector space over the field, with the elements being formal sums:
The algebra structure on the vector space is defined using the multiplication in the group:
where on the left, g and h indicate elements of the group algebra, while the multiplication on the right is the group operation (written as multiplication).
Because the above multiplication can be confusing, one can also write the basis vectors of K as eg (instead of g), in which case the multiplication is written as:
Read more about this topic: Group Ring
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