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On the Primitivity of the Group Algebra

Published online by Cambridge University Press:  20 November 2018

Alan Rosenberg
Alan Rosenberg*
Affiliation:
Wesleyan University, Middletown, Connecticut
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Let G be a group and F a field of arbitrary characteristic. In [4] Kaplansky asks under what conditions is F[G] primitive, where F[G] is the group algebra of G over F. We give some necessary conditions on G that F[G] be primitive and propose a conjecture.

Definition. A ring R is 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 that F[G] is primitive, there are some immediate restrictions on G. First of all G cannot be Abelian since the only primitive commutative rings are fields. (I exclude of course the case when G consists of one element.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 536 - 540
Copyright
Copyright © Canadian Mathematical Society 1971

References

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