Given a group , the center of , denoted as , 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 of is termed central if .
Central subgroups have the following properties:
- They are abelian groups.
- They are normal subgroups. In fact, they are central factors, and are hence transitively normal subgroups.