Centrally indecomposable group

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 defining ingredient::central product of two proper subgroups.
 * 2) Every proper nontrivial defining ingredient::central factor is a defining ingredient::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 $$Z(G)$$.

Stronger properties

 * Simple group

Weaker properties

 * Directly indecomposable group
 * Centerless group