Centre of a group: Difference between revisions
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In [[group theory]], the '''centre of a group''' is the subset of elements which [[commutativity|commute]] with every element of the group. | In [[group theory]], the '''centre of a group''' is the subset of elements which [[commutativity|commute]] with every element of the group. | ||
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:<math>Z(G) = \{ z \in G : \forall x \in G,~xz=zx \} . \,</math> | :<math>Z(G) = \{ z \in G : \forall x \in G,~xz=zx \} . \,</math> | ||
The centre is a [[subgroup]], which is [[normal subgroup|normal]] and indeed [[characteristic subgroup|characteristic]]. It is the [[kernel]] of the [[group homomorphism|homomorphism]] to ''G'' to its [[inner automorphism]] group. | The centre is a [[subgroup]], which is [[normal subgroup|normal]] and indeed [[characteristic subgroup|characteristic]]. It may be described as the set of elements by which [[conjugation (group theory)|conjugation]] is trivial (the identity map); this shows the centre as the [[kernel of a homomorphism|kernel]] of the [[group homomorphism|homomorphism]] to ''G'' to its [[inner automorphism]] group. | ||
==References== | ==References== | ||
* {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=14 }} | * {{cite book | author=Marshall Hall jr | title=The theory of groups | publisher=Macmillan | location=New York | year=1959 | pages=14 }} | ||
Latest revision as of 11:29, 13 February 2009
In group theory, the centre of a group is the subset of elements which commute with every element of the group.
Formally,
The centre is a subgroup, which is normal and indeed characteristic. It may be described as the set of elements by which conjugation is trivial (the identity map); this shows the centre as the kernel of the homomorphism to G to its inner automorphism group.
References
- Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 14.