commutator length

English

commutator length (plural commutator lengths)

The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators.
In ${\\displaystyle \\langle a,b|b^{2}=1\\rangle }$ , the commutator length of ${\\displaystyle aba^{-2}ba}$ is two, since it equals ${\\displaystyle (aba^{-1}b^{-1})(b^{-1}a^{-1}ba)}$ 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 ${\\displaystyle \\langle a,b|b^{2}=1\\rangle }$ is at least two, since there's an element of commutator length two in it.

单词	commutator length
释义	commutator length English Noun commutator length (plural commutator lengths) The number of multiplicands needed, at a minimum, to express a given group element as a product of commutators. In ${\\displaystyle \\langle a,b\|b^{2}=1\\rangle }$ , the commutator length of ${\\displaystyle aba^{-2}ba}$ is two, since it equals ${\\displaystyle (aba^{-1}b^{-1})(b^{-1}a^{-1}ba)}$ 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 ${\\displaystyle \\langle a,b\|b^{2}=1\\rangle }$ is at least two, since there's an element of commutator length two in it.* Derived terms stable commutator length
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