Let d be a metric on the symmetric group Sn. Suppose d is left invariant . i.e, d(ca,cb)=d(a,b) for all a,b,c in Sn.

Is there any method to convert it as a right invariant metric.

i.e, d(ac,bc)=d(a,b) for all a,b,c in Sn ?

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- Aug 3rd 2008, 05:31 AMthipplisymmetric group metric
Let d be a metric on the symmetric group Sn. Suppose d is left invariant . i.e, d(ca,cb)=d(a,b) for all a,b,c in Sn.

Is there any method to convert it as a right invariant metric.

i.e, d(ac,bc)=d(a,b) for all a,b,c in Sn ? - Aug 3rd 2008, 06:16 AMNonCommAlg
let G be any group and $\displaystyle d$ a left invariant metric defined on G. then $\displaystyle \rho$ defined by $\displaystyle \rho(x,y)=d(x^{-1},y^{-1}), \ \forall \ x,y \in G$

is a right invariant metric on G. (very easy to see. so i leave the proof to you!)