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Suppose G is a group, x,y are (possibly equal, possibly distinct) elements of G, and n is an integer. The n-commutator of x,y in G, denoted [x,y]_n, is defined as:

[x,y]_n = (xy)^ny^{-n}x^{-n}

Note that:

  • For n = 0 and n = 1, the n-commutator is always the identity element.
  • For n = -1, the n-commutator is the inverse of a conjugate of the usual commutator [x,y], and for n = 2, the n-commutator is a conjugate of the usual commutator.
  • For any other n, the n-commutator is in the normal subgroup generated by [x,y], but it may generate a strictly smaller normal subgroup.

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