Difference between revisions of "Commutator"
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Latest revision as of 10:24, 12 May 2015
As always, 1 and e will be used to denote the identity of a group.
Definition
Given a group (G,×) we define the commutator of two elements, g,h∈G as:
- [g,h]=ghg−1h−1[1] (I use this definition, as does Serge Lang)
Although some people use:
- [g,h]=g−1h−1gh[2]
I prefer and use the version given by Serge Lang, just because it better aligns with alphabetical order, that is to say that g,h commute is to say gh=hg (which leads to ghg−1h−1=e) and hg=gh while logically equivalent, seems a little bit nastier to write (and leads to hgh−1g−1=e)
Important property
[Expand]
Theorem: The commutator [g,h]=e if and only if the elements g and h commute
Identities
See also
References
- Jump up ↑ Serge Lang - Algebra - Revised Third Edition - GTM
- Jump up ↑ http://en.wikipedia.org/w/index.php?title=Commutator&oldid=660112221#Group_theory
