semisimple group

In group theory the use of the phrase semi-simple group is used sparingly.Standard texts on group theory including [1, 2] avoidthe term altogether. Other texts provide precise definitions which are neverthelessnot equivalent [3, 4]. In general it is preferable to useother terms to describe the class of groups being considered as there isno uniform convention. However, below is a list of possible uses of forthe phrase semi-simple group.

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

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

3. A product of simple groups may be called semi-simple. Dependingon application, the simple groups may be further restricted to finite simple groupsand may also exclude the abelian simple groups.

4. A Lie group whose associated Lie algebra is a semi-simple Lie algebra maybe called a semi-simple group and more specifically, asemi-simple Lie group.

Connections with algebra

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

For example. In a finite dimensional associative algebra $A$, if $A$ it is a product ofsimple algebras then the Jacobson radical is trivial. In contrast, if $A$ has trivialJacobson radical then it is a direct product of simple algebras. Thus $A$ may becalled 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-simpleby virtue of having no non-trivial normal abelian subgroups then $S_n$ is semi-simplefor all $n>5$. However, $S_n$ is not a product of simple groups.