# Centrally indecomposable group

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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

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## 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 .