Centre of a group: Difference between revisions
Jump to navigation
Jump to search
imported>Richard Pinch (def in terms of trivial conjugation) |
mNo edit summary |
||
| (2 intermediate revisions by one other user not shown) | |||
| Line 1: | Line 1: | ||
{{subpages}} | |||
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. | ||
| Line 6: | Line 7: | ||
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. | 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 }}[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:00, 26 July 2024
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.