Centrally indecomposable group

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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

• Simple group

Weaker properties

• Directly indecomposable group
• Centerless group