Semisimple group: Difference between revisions
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{{variationof|simplicity}} | {{variationof|simplicity}} | ||
''The term semisimple has also been used at some places for a group whose [[solvable | ''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== | ==Definition== | ||
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A group is said to be '''semisimple''' if it occurs as a [[central product]] of (possibly more than two) [[quasisimple group]]s. | 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: | 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: | ||
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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== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::simple non-Abelian group]] || || || || | |||
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| [[Weaker than::quasisimple group]] || || || || | |||
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| [[Weaker than::characteristically simple group]] || || || || | |||
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===Weaker properties=== | ===Weaker properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[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 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 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.