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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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 proper nontrivial central factor. However, for an group with a nontrivial center, the center itself is a central factor.
Definition with symbols
A group is said to be a centrally indecomposable group if we cannot write:
viz., as a central product for proper subgroups and of .