On the Primitivity of the Group Algebra

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Abstract
LetGbe a group andFa field of arbitrary characteristic. In [4] Kaplansky asks under what conditions isF[G] primitive, whereF[G] is the group algebra ofGoverF. We give some necessary conditions onGthatF[G] be primitive and propose a conjecture.Definition.A ringRis primitive if it has a faithful irreducible right module.The above should really be considered as a definition of right primitive. One can analogously define left primitive and the two properties are not equivalent. For our purposes, the two concepts are equivalent, for the group algebra possesses a nice involution.If we assume thatF[G] is primitive, there are some immediate restrictions onG. First of allGcannot be Abelian since the only primitive commutative rings are fields. (I exclude of course the case whenGconsists of one element.)

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