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 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

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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