# symmetric group metric

• Aug 3rd 2008, 05:31 AM
thippli
symmetric 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 AM
NonCommAlg
Quote:

Originally Posted by thippli
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 ?

let G be any group and $d$ a left invariant metric defined on G. then $\rho$ defined by $\rho(x,y)=d(x^{-1},y^{-1}), \ \forall \ x,y \in G$

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