Commutator (group theory)
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

This is an excerpt from the article Commutator (group theory) from the Wikipedia free encyclopedia. A list of authors is available at Wikipedia.
The article Commutator (group theory) at en.wikipedia.org was accessed 14 times in the last 30 days. (as of: 05/23/2014)
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Commutator - Wikipedia, the free encyclopedia
. The commutator of two elements, g and h, of a group G, is the element. [g, h ] = g−1h−1gh. It is equal to the group's identity if ...
en.wikipedia.org/wiki/Commutator
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Commutator subgroup - Wikipedia, the free encyclopedia
The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this ... 2.1 Derived series; 2.2 Abelianization; 2.3 Classes of groups .... Group theory · Functional subgroups.
en.wikipedia.org/wiki/Commutator_subgroup
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Commutator -- from Wolfram MathWorld
There is a related notion of commutator in the theory of groups. The commutator of two group elements A and B is ABA^(-1)B^(-1) , and two elements A and B ...
mathworld.wolfram.com/Commutator.html
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Commutator (group theory) - The Free Dictionary
com·mu·ta·tor (k m y -t t r). n. 1. A cylindrical arrangement of insulated metal bars connected to the coils of a direct-current electric motor or generator, providing a ...
www.thefreedictionary.com/Commutator+(group+theory)
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group theory - Let $C$ be the commutator subgroup of $G$. Prove ...
Nov 30, 2011 ... This can be done for any elements, so $G/C$ is abelian. This seems somewhat surprisingly short. Is that all there is to it? Regards. group-theory ...
math.stackexchange.com/questions/87082/
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Commutator Subgroup - Group Theory - MathOverflow
Jul 9, 2012 ... The commutator subgroup of the free group $\langle a,b \rangle$ is freely generated by the set \lbrace [a^n,b^m] \mid n,m \in \mathbb Z, nm ...
mathoverflow.net/questions/101746/commutator-subgroup-group-theory
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gr.group theory - Commutator subgroup does not consist only of ...
Oct 30, 2010 ... The problem is whether the commutator subgroup may contain elements that are not commutators. One example are the free groups.
mathoverflow.net/questions/44269/
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gr.group theory - Non-commutator in simple group? - MathOverflow
May 1, 2012 ... For a group $G$, we say that $x\in G$ is a commutator if there exists $a,b \in G$ such that $x=a^{1}b^{-1}ab$, and we say that $x$ is a ...
mathoverflow.net/questions/95692/non-commutator-in-simple-group
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On commutators in groups - University of Evansville
characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their  ...
faculty.evansville.edu/rm43/publications/commutatorsurvey.pdf
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COMMUTATOR THEORY FOR LOOPS 1 ... - Math Department
sults summarized in Corollary 5.2. Generating sets in terms of elementwise associators and commutators, a traditional approach of group theory and loop theory, ...
web.cs.du.edu/~petr/data/papers/commutator_theory_for_loops.pdf
Search results for "Commutator (group theory)"
Commutator (group theory) in science
Review of Group Theory: The Commutator Subgroup and - Abstract ...
Feb 27, 2011 ... Motivation Very often in group theory we are dealt groups which, to our… ... In essence the commutator subgroup is precisely such that when the ambient .... Pingback by University of Maryland College Park Qualifying Exams ...
[PDF]On commutators in groups - University of Evansville
characters of nonabelian groups. They are now an established and immensely useful tool in all of group theory. Commutators became objects of interest in their  ...
[PDF]GROUP THEORY NOTES FOR THE COURSE ... - McGill University
MCGILL UNIVERSITY, FALL 2003, ... Commutator subgroup ... at least in the theory of finite groups on which this course focuses, there is no comparable theory ...
[PDF]Commutator Theory for Loops
joint work with Petr Vojtechovský, University of Denver ... Commutators for loops. 1 / 11 ... Bruck's approach (1950's), by direct analogy to group theory:.
University of Isfahan International Journal of Group Theory 2251 ...
University of Isfahan International Journal of Group Theory 2251-7650 3 1 2014 ... (soluble-by-finite) group whose proper subgroups have minimax commutator ...
Commutator - Wikipedia, the free encyclopedia
1 Group theory. 1.1 Identities. 2 Ring theory. 2.1 Identities. 3 Graded rings and algebras; 4 Derivations; 5 Anticommutator; 6 See also; 7 Notes; 8 References ...
[PDF]Commutator Theory for Loops
Commutator Theory for Loops. David Stanovský. Charles University, Prague, Czech Republic stanovsk@karlin.mff.cuni.cz joint work with Petr Vojtechovský, ...
Stable commutator length on mapping class groups
Jun 11, 2013 ... Mathematics > Group Theory ... the Nielsen-Thurston decomposition) when an element $g\in\Gamma$ has positive stable commutator length.
Books on the term Commutator (group theory)
Commutator Calculus and Groups of Homotopy Classes
Hans Joachim Baues, 1981
PART A: HOMOTOPY OPERATIONS, NILPOTENT GROUP THEffl AND NILPOTENT LIE ALGEBRA THEORY L COMMUTATOR CALCULUS Commutator calculus is a branch of group theory dealing with nilpotent groups, Lie algebras, the ...
Group Theory: An Intuitive Approach
R. Mirman, 1995
What is the commutator subgroup of the quaternion group (pb. III.2.e-4, p. 74)? Find the commutator subgroups of the geometric groups, and discuss the reasons, group theoretical and particularly geometrical, for elements being commutators, ...
Combinatorial Group Theory, Discrete Groups, and Number ...
Benjamin Fine, Anthony M. Gaglione, Dennis Spellman, 2006
Introduction This paper addresses some questions concerning the commutator calculus - an important area of Combinatorial Group Theory. We first recall some basic definitions and terminology. Suppose G is a group. Let a,b £ G. Then the ...
Fundamentals of Group Theory: An Advanced Approach
Steven Roman, 2011
Nontrivial groups with no nontrivial proper normal subgroups are, in some sense, simple. ... note that authors who define the conjugate of 84 Fundamentals of Group Theory Simple Groups Commutators of Elements Commutators of Elements.

Blog posts on the term
Commutator (group theory)
You can solve the cube – with commutators! | Geometry and the imagination
You can solve the cube – with commutators! August 24, 2013 in Groups | Tags: commutators, group theory, Rubik's cube After a couple years of living out of suitcases, we recently sold our house in Pasadena, and bought a new one in Hyde Park. All our junk was shipped to us, and the boxes we didn’t feel like unpacking are all sitting around in the attic, where the kids have been spending a lot of time this summer.
lamington.wordpress.com/2013/08/24/you-can-solve-the-cube-with-commutators/
Review of Group Theory: The Commutator Subgroup and the Abelianization of a Group « Abstract Nonsense
Point of post: In this post we discuss the notion of the commutator subgroup of a group $latex G$ and prove some basic, yet important, facts about it. We then discuss the abelianization of a group (the group mod the commutator subgroup). Motivation Very often in group theory we are dealt groups which, to our…
drexel28.wordpress.com/2011/02/27/review-of-group-theory-the-commutator-subgroup-and-the-abelianization-of-a-group/
Operator Algebra II – The Commutator | Mathematics in Chemical Theory
Anybody who has taken an undergraduate course on quantum chemistry or quantum mechanics is familiar with the commutator on some level. In these courses, I was told how to use it to carry out some basic (but very important!) calculations, and was given a teaser of its physical significance in terms of incompatible observables. In…
chemicalmathematics.wordpress.com/2013/02/06/operator-algebra-ii-the-commutator/
Perfect groups of infinite commutator length: setup | On music, computing and math
At the end of my post Some trivia on derived subgroups I quickly mention that what I had been discussing up to then is essentially about the commutator length of the group. Recall that the derived subgroup is the group generated by all commutators, and that a group is said to be perfect if and only…
pbelmans.wordpress.com/2012/09/11/perfect-groups-of-infinite-commutator-length-setup/
Conjugacy class of commutators of generators for a free group. | | Free EducationFree Education
]]> Let $F$ be a free group generated by two elements. Let $\{a,b\}$ and $\{c,d\}$ are two different generating set.
3mr.me/conjugacy-class-of-commutators-of-generators-for-a-free-group/
(Collective) Introduction to Commutators
This is more experimental than most threads, and I've been hoping to do a collective tutorial at some point. Let's talk about Comms. How would we go about teaching this very very important subject to noobs, with everything they could need to know? There have been a few tutorials floating about over the years, and quite a few explanations. I think there needs to finally be a "from noob to optimizer" walkthrough of sorts that can finally bridge this gap. I won't even contribute right
If we have a group $latex G$, it may or may not be abelian. We can measure how nonabelian it is with the commutator subgroup $latex \left[G,G\right]$. This is the subgroup generated by all the commutators in $latex G$. Start with all the elements of $latex G$ of the form $latex ghg^{-1}h^{-1}$ with $latex g$…