# Difference between revisions of "Centrally indecomposable group"

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{{semistddef}} | {{semistddef}} | ||

{{group property}} | {{group property}} | ||

− | {{ | + | {{variation of|simple group}} |

==Definition== | ==Definition== | ||

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===Symbol-free definition=== | ===Symbol-free definition=== | ||

− | A nontrivial [[group]] is said to be '''centrally indecomposable''' if it | + | A nontrivial [[group]] is said to be '''centrally indecomposable''' if it satisfies the following equivalent conditions: |

− | + | # It cannot be expressed as the [[defining ingredient::central product]] of two proper subgroups. | |

+ | # 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 | + | 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> | + | # 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>. | |

− | viz., as a [[central product]] for proper subgroups <math>H</math> and <math>K</math> of <math>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

View a complete list of group propertiesVIEW 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 satisfies the following equivalent conditions:

- It cannot be expressed as the central product of two proper subgroups.
- 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 is said to be a centrally indecomposable group if it satisfies the following equivalent conditions:

- We cannot write , viz., as a central product for proper subgroups and of .
- Every proper nontrivial central factor of is a central subgroup of , i.e., it is contained in the center .