• STUDY OF JACOBSON RADICAL OF THE GROUP ALGEBRA OF A GROUP IS EITHER COMMUTATIVE OR CENTRAL SUB ALGEBRA

Abstract

This study constructed the relationship between Radical and Group Algebra. In order to construct the relationship between radical and group algebra let a group algebra K (G) of a finite group G over the field K of characteristic p has a nonzero radical R if and only if p is a divisor of o(G), the order of G. This study shows that the appearances were not deceptive in the problem of centrality, for odd primes and in the problem of commutativity. Finally after introducing commutativity of Ring Theory and using some Theorems and Lemmas we have verified that Jacobson radical of the Group Algebra of a Group is either commutative or a central sub algebra.

Key definition: Group, Finite group, Subgroup, Abelian group, Commutator, Sylow p-subgroup, Homomorphism, Isomorphism.