Difference between revisions of "Centrally indecomposable group"

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(Definition)
(Definition)
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A nontrivial [[group]] is said to be '''centrally indecomposable''' if it cannot be expressed as the central product of two proper subgroups.
 
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 non-abelian group, this is equivalent to saying that there is no nontrivial [[central factor]]. However, for an abelian group, this is not equivalent, because, for instance, a [[cyclic group]] of order <math>p^2</math> is centrally indecomposable but it has a proper nontrivial central factor.
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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===
 
===Definition with symbols===

Revision as of 01:12, 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.

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