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 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 $H \leq Z(G)$.

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.

References

• Hazewinkel, Michiel, ed. (2001), "Centre of a group", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=C/c021250 .