Difference between revisions of "Centrally indecomposable group"

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viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>.
 
viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>.
 
==Formalisms==
 
 
{{obtainedbyapplyingthe|simple group operator|central factor}}
 
 
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==
 
==Relation with other properties==

Revision as of 01:13, 13 September 2009

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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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Definition

Symbol-free definition

A nontrivial group is said to be centrally indecomposable if it cannot be expressed as the central product of two proper subgroups.

Note that, for a centerless group, this is equivalent to saying that there is no nontrivial central factor. However, for an group with a nontrivial center, the center itself is a central factor.

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 proper subgroups H and K of G.

Relation with other properties

Stronger properties

Weaker properties