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 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 semi-simple group.

1.

A group is semi-simple if it has no non-trivial normal abelian subgroups [3, p. 89].
2.

A group $G$ is semi-simple if $G^{\prime}=G$ and $G/Z(G)$ is a direct product of non-abelian simple groups [4, Def. 6.1].
3.

A product 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.

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.

Connections with algebra

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 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 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 $S_{n}$ is semi-simple 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) 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