# Difference between revisions of "Semisimple group"

(→Definition with symbols) |
(→Relation with other properties) |
||

Line 23: | Line 23: | ||

===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

+ | |- | ||

+ | | [[Weaker than::simple non-Abelian group]] || || || || | ||

+ | |- | ||

+ | | [[Weaker than::quasisimple group]] || || || || | ||

+ | |- | ||

+ | | [[Weaker than::characteristically simple group]] || || || || | ||

+ | |} | ||

===Weaker properties=== | ===Weaker properties=== | ||

− | + | {| class="sortable" border="1" | |

+ | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

+ | |- | ||

+ | | [[Stronger than::perfect group]] || equals its own [[derived subgroup]] || [[semisimple implies perfect]] || [[perfect not implies semisimple]] || | ||

+ | |} | ||

==Metaproperties== | ==Metaproperties== |

## Latest revision as of 19:29, 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

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.