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
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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 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 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.

Relation with other properties

Stronger properties

Weaker properties