Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-13T08:16:12.607Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Connell, I., On the group ring, Can. J. Math. 15 (1963), 650–685.Google Scholar
2. Herstein, I. N., Noncommutative rings (Carus Monograph No. 15, MAA, 1968).Google Scholar
3. Jacobson, N., Structure of rings (Amer. Math. Soc. Coll., vol. 37, rev. éd., 1964).Google Scholar
4. Kaplansky, I., Problems in the theory of rings revisited, Amer. Math. Monthly 77 (1970), 445–454.Google Scholar
5. Kaplansky, I., Groups with representation of bounded degree, Can. J. Math. 1 (1949), 105–112.Google Scholar
6. Rosenberg, A. E., On the semisimplicity of the group algebra (Thesis, Brown University, Providence, R.I., 1969).Google Scholar
7. Scott, W. R., Group theory (Prentice Hall, New Jersey, 1964).Google Scholar