Central subgroup

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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 and is also an abelian group (because, in particular, all elements of the center must commute with each other). A subgroup H of G 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.