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 ${\displaystyle G}$, the center of ${\displaystyle G}$, denoted as ${\displaystyle Z(G)}$, 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 and is also an abelian group (because, in particular, all elements of the center must commute with each other). A subgroup ${\displaystyle H}$ of ${\displaystyle G}$ is termed central if ${\displaystyle H\leq Z(G)}$.

Central subgroups have the following properties:

• They are abelian groups.
• They are normal subgroups. They are central factors, and are hence transitively normal subgroups.