# Difference between revisions of "Centrally indecomposable group"

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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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.
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## 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 non-abelian group, this is equivalent to saying that there is no nontrivial central factor. However, for an abelian group, this is not equivalent, because, for instance, a cyclic group of order $p^2$ is centrally indecomposable but it has a proper nontrivial central factor.

### Definition with symbols

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

$G = H * K$

viz., as a central product for proper subgroups $H$ and $K$ of $G$.

## 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.