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In mathematical field of geometric group theory, a length function is a function that assigns a number to each element of a group.
A length function L : G → R+ on a group G is a function satisfying:
Coxeter groups (including the symmetric group) have combinatorial important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: length of a Weyl group element.
A longest element of a Coxeter group is both important and unique up to conjugation (up to different choice of simple reflections).