Centre of a group: Difference between revisions
Jump to navigation
Jump to search
imported>Richard Pinch (new entry, just a stub) |
imported>Richard Pinch (kernel of morphism to inner automorphism group) |
||
| Line 5: | Line 5: | ||
:<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]]. | 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. | ||
==See also== | ==See also== | ||
Revision as of 10:57, 15 November 2008
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 is the kernel of the homomorphism to G to its inner automorphism group.
See also
References
- Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 14.