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In [[mathematics]], in the field of [[group theory]], a [[subgroup]] of a [[group (mathematics)|group]] is termed '''central''' if it lies inside the center of the group. | |||
Given a group <math>G</math>, the [[center (group theory)|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]] and is also an [[abelian group]] (because, in particular, all elements of the center must commute with each other). 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. | |||
* They are [[normal subgroup]]s. In fact, they are [[central factor]]s, and are hence [[transitively normal subgroup]]s. | |||
== References == | |||
* {{springer|id=C/c021250|title=Centre of a group}}. | |||
[[Category:Subgroup properties]] | |||
Revision as of 09:01, 9 May 2013
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. In fact, they are central factors, and are hence transitively normal subgroups.
References
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