centralizer
Let be a group. The centralizer of an element is defined to be the set
Observe that, by definition, , and that if , then , so that . Thus is a subgroup of . For , the subgroup is non-trivial, containing at least .
To illustrate an application of this concept we prove the following lemma.
Lemma:
There exists a bijection between the right cosets of and the conjugates of .
Proof:
If are in the same right coset, then for some . Thus .
Conversely, if then and giving are in the same right coset.
Let denote the conjugacy class of . It follows that and .
We remark that , where denotes the center of .
Now let be a -group, i.e. a finite group of order , where is a prime and is a positive integer. Let . Summing over elements in distinct conjugacy classes, we have since the center consists precisely of the conjugacy classes of cardinality . But , so . However, is certainly non-empty, so we conclude that every -group has a non-trivial center.
The groups and , for any , are isomorphic.
Title | centralizer |
---|---|
Canonical name | Centralizer |
Date of creation | 2013-03-22 12:35:01 |
Last modified on | 2013-03-22 12:35:01 |
Owner | drini (3) |
Last modified by | drini (3) |
Numerical id | 14 |
Author | drini (3) |
Entry type | Definition |
Classification | msc 20-00 |
Synonym | centraliser |
Related topic | Normalizer |
Related topic | GroupCentre |
Related topic | ClassEquationTheorem |