# Difference between revisions of "Centrally indecomposable group"

From Groupprops

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

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<math>G = H * K</math> | <math>G = H * K</math> | ||

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

==Formalisms== | ==Formalisms== |

## Revision as of 00:18, 29 May 2009

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 simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

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.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Centrally indecomposable group, all facts related to Centrally indecomposable group) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

## 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 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 is centrally indecomposable but it has a proper nontrivial central factor.

### Definition with symbols

A group is said to be a centrally indecomposable group if we cannot write:

viz., as a central product for proper subgroups and of .

## 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.