semisimple group
In group theory the use of the phrase semisimple 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 equivalent^{} [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 semisimple group.

1.
A group is semisimple if it has no nontrivial normal abelian^{} subgroups^{} [3, p. 89].

2.
A group $G$ is semisimple if ${G}^{\prime}=G$ and $G/Z(G)$ is a direct product^{} of nonabelian^{} simple groups^{} [4, Def. 6.1].

3.
A product^{} of simple groups may be called semisimple. Depending on application, the simple groups may be further restricted to finite simple groups and may also exclude the abelian simple groups.

4.
A Lie group whose associated Lie algebra is a semisimple Lie algebra may be called a semisimple group and more specifically, a semisimple Lie group.
Connections with algebra^{}
The use of semisimple 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 radical^{} is trivial. In contrast, if $A$ has trivial Jacobson radical then it is a direct product of simple algebras. Thus $A$ may be called semisimple 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 semisimple by virtue of having no nontrivial normal abelian subgroups then ${S}_{n}$ is semisimple for all $n>5$. However, ${S}_{n}$ is not a product of simple groups.
References
 1 Aschbacher, M. Finite group^{} 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) Springerverlag, Berlin (1982, 1986).
Title  semisimple group 

Canonical name  SemisimpleGroup 
Date of creation  20130322 13:17:07 
Last modified on  20130322 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  semisimple group 
Defines  semisimple group 