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 , 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. They are central factors, and are hence transitively normal subgroups.

References

• Hazewinkel, Michiel, ed. (2001), "Centre of a group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 .