Semisimple group: Difference between revisions
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{{group property}} | |||
{{variationof|simplicity}} | |||
==Definition== | |||
===Symbol-free definition=== | |||
A group is said to be '''semisimple''' if it occurs as a [[central product]] of (possibly more than two) [[quasisimple group]]s. | |||
====Definition with symbols=== | |||
A group <math>G</math> is said to be semisimple if there are subgroups <math>G_1, G_2, \ldots, G_r</math> such that: | |||
* Each <math>G_i</math> is quasisimple | |||
* The <math>G_i</math>s generate <math>G</math> | |||
* The group <math>[G_i, G_j]</math> is trivial | |||
==Relation with other properties== | |||
===Stronger properties=== | |||
* [[Simple non-Abelian group]] | |||
* [[Quasisimple group]] | |||
===Weaker properties=== | |||
* [[Perfect group]] | |||
==Metaproperties== | |||
{{S-universal}} | |||
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. | |||
{{Q-closed}} | |||
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. | |||
{{DP-closed}} | |||
A direct product of semisimple groups is semisimple. In fact, any central product of semisimple groups is semisimple. | |||
Revision as of 05:43, 31 March 2007
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
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
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.