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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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


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.

Relation with other properties

Stronger properties

Weaker properties