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 [math]\displaystyle{ G }[/math], the center of [math]\displaystyle{ G }[/math], denoted as [math]\displaystyle{ Z(G) }[/math], 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. A subgroup [math]\displaystyle{ H }[/math] of [math]\displaystyle{ G }[/math] is termed central if [math]\displaystyle{ H \leq Z(G) }[/math].

Central subgroups have the following properties:

• They are abelian groups (because, in particular, all elements of the center must commute with each other).
• 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 Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=C/c021250 .