Thread: 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 ?

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 a left invariant metric defined on G. then defined by

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