The center of a group G is the subgroup consisting of those elements that commute with every other element. Formally,
It can be shown that the center has the following properties:
It is a normal subgroup (in fact, a characteristic subgroup).
It consists of those conjugacy classes containing just one element.
The center of an abelian group is the entire group.
For every prime p , every non-trivial finite p -group ( http://planetmath.org/PGroup4 ) has a non-trivial center. (Proof of a stronger version of this theorem. ( http://planetmath.org/ProofOfANontrivialNormalSubgroupOfAFinitePGroupGAndTheCenterOfGHaveNontrivialIntersection ))
A subgroup of the center of a group G is called a central subgroup of G . All central subgroups of G are normal in G .
For any group G , the quotient ( http://planetmath.org/QuotientGroup ) G / Z ( G ) is called the central quotient of G , and is isomorphic to the inner automorphism group Inn ( G ) .
| Title | center of a group |
|---|---|
| Canonical name | CenterOfAGroup |
| Date of creation | 2013-03-22 12:23:38 |
| Last modified on | 2013-03-22 12:23:38 |
| Owner | yark (2760) |
| Last modified by | yark (2760) |
| Numerical id | 20 |
| Author | yark (2760) |
| Entry type | Definition |
| Classification | msc 20A05 |
| Synonym | center |
| Synonym | centre |
| Related topic | CenterOfARing |
| Related topic | Centralizer |
| Defines | central quotient |
