Central subgroup

In mathematics, in the field of group theory, a subgroup of a group is termed central if it lies inside the center of the group.

Given a group $$G$$, the center of $$G$$, denoted as $$Z(G)$$, is defined as the set of those elements of the group which commute with every element of the group. The center is a characteristic subgroup. A subgroup $$H$$ of $$G$$ is termed central if $$H \leq Z(G)$$.

Central subgroups have the following properties:


 * They are abelian groups (because, in particular, all elements of the center must commute with each other).
 * They are normal subgroups. They are central factors, and are hence transitively normal subgroups.