Centrally indecomposable group
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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 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 is centrally indecomposable but it has a proper nontrivial 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 .
In terms of the simple group operator
This property is obtained by applying the simple group operator to the property: central factor
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