Centrally indecomposable group: Difference between revisions

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{{semistddef}}
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{{variation of|simple group}}
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==Definition==
==Definition==
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A nontrivial [[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 cannot be expressed as the [[defining ingredient::central product]] of two proper subgroups.
* 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.
# Every proper nontrivial [[defining ingredient::central factor]] is a [[defining ingredient::central subgroup]], i.e., it is contained in the [[center]] of the group.


===Definition with symbols===
===Definition with symbols===


A [[group]] <math>G</math> is said to be a centrally indecomposable group if we cannot write:
A [[group]] <math>G</math> is said to be a centrally indecomposable group if it satisfies the following equivalent conditions:
 
<math>G = H * K</math>
 
viz., as a [[central product]] for nontrivial groups <math>H</math> and <math>K</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]].
# We cannot write <math>G = H * K</math>, viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>G</math>.
# Every proper nontrivial [[central factor]] of <math>G</math> is a [[central subgroup]] of <math>G</math>, i.e., it is contained in the [[center]] <math>Z(G)</math>.


==Relation with other properties==
==Relation with other properties==

Latest revision as of 01:49, 13 September 2009

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

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 is a variation of simple group|Find other variations of simple group |

Definition

Symbol-free definition

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

  1. It cannot be expressed as the central product of two proper subgroups.
  2. Every proper nontrivial central factor is a central subgroup, i.e., it is contained in the center of the group.

Definition with symbols

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

  1. We cannot write G=H*K, viz., as a central product for proper subgroups H and K of G.
  2. Every proper nontrivial central factor of G is a central subgroup of G, i.e., it is contained in the center Z(G).

Relation with other properties

Stronger properties

Weaker properties

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