The commutator length 
of an element 
in the commutator subgroup of a group 
is the smallest number of commutators
whose product is 
. Thus,
![cl_(G)(g)=min{m:g=[a_1,b_1]...[a_m,b_m] for a_i,b_i in G},](/images/equations/CommutatorLength/NumberedEquation1.svg) |
where ![[a,b]=aba^(-1)b^(-1)](/images/equations/CommutatorLength/Inline5.svg)
.
By convention, 
when 
does not belong to the commutator subgroup
(Calegari 2009).
Commutator Length
See also
Commutator, Commutator Subgroup, Stable Commutator LengthExplore with Wolfram|Alpha
References
Calegari, D. scl Tokyo, Japan: Mathematical Society of Japan, pp. 13-14, 2009. https://doi.org/10.2969/msjmemoirs/020010000.Cite this as:
Weisstein, Eric W. "Commutator Length." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CommutatorLength.html