# Semisimple group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

The term semisimple has also been used at some places for a group whose solvable radical is trivial, which is equivalent to being a Fitting-free group

## Definition

### Symbol-free definition

A group is said to be semisimple if it occurs as a central product of (possibly more than two) quasisimple groups.

### Definition with symbols

A group $G$ is said to be semisimple if there are subgroups $G_1, G_2, \ldots, G_r$ such that:

• Each $G_i$ is quasisimple
• The $G_i$s generate $G$
• The group $[G_i, G_j]$ is trivial for all $i \ne j$

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
simple non-Abelian group
quasisimple group
characteristically simple group

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
perfect group equals its own derived subgroup semisimple implies perfect perfect not implies semisimple

## Metaproperties

Every finite group can be realized as a subgroup of a semisimple group. This follows from the fact that every finite group can be realized as a subgroup of a simple non-Abelian group.

### Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Every quotient of a semisimple group is semisimple. This follows from the fact that every quotient of a quasisimple group is quasisimple, and that a central product is preserved on going to the quotient.

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of semisimple groups is semisimple. In fact, any central product of semisimple groups is semisimple.