semisimple group

In group theory the use of the phrase semi-simple group is used sparingly. Standard texts on group theory including [1, 2] avoid the term altogether. Other texts provide precise definitions which are nevertheless not equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath [3, 4]. In general it is preferable to use other terms to describe the class of groups being considered as there is no uniform convention. However, below is a list of possible uses of for the phrase semi-simple group.

  1. 1.

    A group is semi-simple if it has no non-trivial normal abelianMathworldPlanetmath subgroupsMathworldPlanetmathPlanetmath [3, p. 89].

  2. 2.

    A group G is semi-simple if G=G and G/Z(G) is a direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of non-abelianMathworldPlanetmathPlanetmath simple groupsMathworldPlanetmathPlanetmath [4, Def. 6.1].

  3. 3.

    A productMathworldPlanetmathPlanetmath of simple groups may be called semi-simple. Depending on application, the simple groups may be further restricted to finite simple groups and may also exclude the abelian simple groups.

  4. 4.

    A Lie group whose associated Lie algebra is a semi-simple Lie algebra may be called a semi-simple group and more specifically, a semi-simple Lie group.

The use of semi-simple in the study of algebras, representation theory, and modules is far more precise owing to the fact that the various possible definitions are generally equivalent.

For example. In a finite dimensional associative algebra A, if A it is a product of simple algebras then the Jacobson radicalMathworldPlanetmath is trivial. In contrast, if A has trivial Jacobson radical then it is a direct product of simple algebras. Thus A may be called semi-simple if either: A is a direct product of simple algebras or A has trivial Jacobson radical.

The analogue fails for groups. For instance. If a group is defined as semi-simple by virtue of having no non-trivial normal abelian subgroups then Sn is semi-simple for all n>5. However, Sn is not a product of simple groups.


  • 1 Aschbacher, M. Finite groupMathworldPlanetmath theory Cambridge studies in advanced mathematics 10, Cambridge University Press, Cambridge, (1986).
  • 2 Gorenstein, D. Finite groups Chelsea Publishing Company, New York, (1980).
  • 3 Robinson, D. J.S. A course in the theory of groups Ed. 2, GTM 80, Springer, New York, (1996).
  • 4 Suzuki, M. Group Theory I,II, (English) Springer-verlag, Berlin (1982, 1986).
Title semisimple group
Canonical name SemisimpleGroup
Date of creation 2013-03-22 13:17:07
Last modified on 2013-03-22 13:17:07
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 6
Author Algeboy (12884)
Entry type Definition
Classification msc 20D05
Related topic socle
Defines semi-simple group
Defines semisimple group