Difference between revisions of "Centrally indecomposable group"

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===Symbol-free definition===
 
===Symbol-free definition===
  
A [[group]] is said to be '''centrally indecomposable''' if it satisfies the following equivalent conditions:
+
A nontrivial [[group]] is said to be '''centrally indecomposable''' if it satisfies the following equivalent conditions:
  
 
* It has no proper nontrivial [[central factor]]
 
* It has no proper nontrivial [[central factor]]
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<math>G = H * K</math>
 
<math>G = H * K</math>
  
viz, a [[central product]] for nontrivial groups <math>H</math> and <math>K</math>.
+
viz., as a [[central product]] for nontrivial groups <math>H</math> and <math>K</math>.
  
 
==Formalisms==
 
==Formalisms==

Revision as of 19:49, 14 October 2008

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
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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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View a list of other standard non-basic definitions

Definition

Symbol-free definition

A nontrivial group is said to be centrally indecomposable if it satisfies the following equivalent conditions:

  • It has no proper nontrivial central factor
  • It cannot be expressed as the central product of two proper subgroups, or equivalently, for any proper subgroup, the product with its centralizer is again proper.

Definition with symbols

A group G is said to be a centrally indecomposable group if we cannot write:

G = H * K

viz., as a central product for nontrivial groups H and K.

Formalisms

In terms of the simple group operator

This property is obtained by applying the simple group operator to the property: central factor
View other properties obtained by applying the simple group operator

The group property of being centrally indecomposable is obtained by applying the simple group operator to the subgroup property of being a central factor.

Relation with other properties

Stronger properties

Weaker properties