# Centrally indecomposable group

From Groupprops

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 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 .