Difference between revisions of "Centrally indecomposable group"

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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 [[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.
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Note that, for a [[centerless group]], this is equivalent to saying that there is no proper 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 01:47, 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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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 simple group|Find other variations of simple group |

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 proper 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