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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Definition

Symbol-free definition

A nontrivial group is said to be centrally indecomposable if it satisfies the following equivalent conditions:

  1. It cannot be expressed as the central product of two proper subgroups.
  2. Every proper nontrivial central factor is a central subgroup, i.e., it is contained in the center of the group.

Definition with symbols

A group G is said to be a centrally indecomposable group if it satisfies the following equivalent conditions:

  1. We cannot write G = H * K, viz., as a central product for proper subgroups H and K of G.
  2. Every proper nontrivial central factor of G is a central subgroup of G, i.e., it is contained in the center Z(G).

Relation with other properties

Stronger properties

Weaker properties