Group Algebras

Abstract

Let G be a finite group. We denote by Z[G] the set of all formal linear combinations of elements of G with integer coefficients. In Z[G] we define the operations of addition and multiplication in a quite natural way:

$$ \left. \begin{gathered} \mathop{\sum }\limits_{i} {{n}_{i}}{{g}_{i}} + \mathop{\sum }\limits_{i} {{m}_{i}}{{g}_{i}} = \mathop{\sum }\limits_{i} \left( {{{n}_{i}} + {{m}_{i}}} \right){{g}_{i}}, \hfill \\ \mathop{\sum }\limits_{i} {{n}_{i}}{{g}_{i}}\cdot \mathop{\sum }\limits_{j} {{m}_{j}}{{g}_{j}} = \mathop{\sum }\limits_{{i,j}} {{n}_{i}}{{m}_{j}}{{g}_{i}}{{g}_{j}}. \hfill \\ \end{gathered} \right\} $$

((1))

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