Let P_{n} be group of all permutations of the set

Mary Reyes

Mary Reyes

Answered question

2022-01-21

Let Pn be group of all permutations of the set {1,,n}. I have trouble with construction a Lipschitz function f:PnR.

Answer & Explanation

Heather Fulton

Heather Fulton

Beginner2022-01-21Added 31 answers

Every function f on a finite metric space (X,d) is Lipschitz.
To wit, call D the minimal distance between distinct points of X, and M the diameter of the range of f, that is, the maximal difference between the images by f of different points of X.
Then, |f(x)f(y)|(MD)d(x,y) for every x and y in X hence f is Lipschitz on X with constant M/D.
encolatgehu

encolatgehu

Beginner2022-01-22Added 27 answers

dn is indeed a metric, and we know that if (X,d) is a metric space, for each x0X the map xd(x,x0) is Lipschitz of constant 1 (it's a consequence of triangular inequality). So for example you can take f(σ)=1ncard(i:σ(i)i), or for a fixed permutation σ0:
fσ0(σ)=1ncard(i:σ(i)σ0(i)).

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