# 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 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

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### In terms of 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.