# Difference between revisions of "Centrally indecomposable group"

From Groupprops

(→Definition) |
(→Definition) |
||

Line 11: | Line 11: | ||

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 | + | 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

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

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |

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 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 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.