Commutator length

English

Noun

1. The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators.

   In $⟨a,b|b^2=1⟩$, the commutator length of $ab{a}^{-2}ba$ is two, since it equals $(ab{a}^{-1}{b}^{-1})(b^{-1}{a}^{-1}ba)$ but isn't a commutator.

2. The supremum, over all elements of a given group's derived subgroup, of their commutator lengths.

   The commutator length of $⟨a,b|b^2=1⟩$ is at least two, since there's an element of commutator length two in it.

Derived terms

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