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>G</math>, the center of <math>G</math>, denoted as <math>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>H</math> of <math>G</math> is termed ''central'' if <math>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 == * {{springer|id=C/c021250|title=Centre of a group}}.

Category:Subgroup properties