# Difference between revisions of "Semisimple group"

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* Each <math>G_i</math> is quasisimple | * Each <math>G_i</math> is quasisimple | ||

* The <math>G_i</math>s generate <math>G</math> | * The <math>G_i</math>s generate <math>G</math> | ||

− | * The group <math>[G_i, G_j]</math> is trivial | + | * The group <math>[G_i, G_j]</math> is trivial for all <math>i \ne j</math> |

==Relation with other properties== | ==Relation with other properties== |

## Revision as of 19:22, 22 May 2012

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 propertiesVIEW 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*

## Contents

## 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 is said to be semisimple if there are subgroups such that:

- Each is quasisimple
- The s generate
- The group is trivial for all

## Relation with other properties

### Stronger properties

### Weaker properties

## 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.