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A [[longest element of a Coxeter group]] is both important and unique up to conjugation (up to different choice of simple reflections).
==Properties==
A group with a length function does ''not'' form a [[filtered group]], meaning that the [[sublevel set]]s <math>S_i := \{g \mid l(g) \leq i\}</math> do not form subgroups in general.
However, the [[group algebra]] of a group with a length functions forms a [[filtered algebra]]: the axiom <math>l(gh) \leq l(g)+l(h)</math> corresponds to the filtration axiom.
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