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.
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