# Centrally indecomposable group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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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 has no proper nontrivial central factor
- It cannot be expressed as the central product of two proper subgroups, or equivalently, for any proper subgroup, the product with its centralizer is again proper.

### Definition with symbols

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

viz., as a central product for nontrivial groups and .

## 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.