commutator length
Noun
Meaning
The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators.
In
⟨
a
,
b
|
b
2
=
1
⟩
{\displaystyle \langle a,b|b^{2}=1\rangle }
, the commutator length of
a
b
a
−
2
b
a
{\displaystyle aba^{-2}ba}
is two, since it equals
(
a
b
a
−
1
b
−
1
)
(
b
−
1
a
−
1
b
a
)
{\displaystyle (aba^{-1}b^{-1})(b^{-1}a^{-1}ba)}
but isn't a commutator.
The supremum, over all elements of a given group's derived subgroup, of their commutator lengths.
The commutator length of
⟨
a
,
b
|
b
2
=
1
⟩
{\displaystyle \langle a,b|b^{2}=1\rangle }
is at least two, since there's an element of commutator length two in it.