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Math Help - symmetric group metric

  1. #1
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    Question 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 ?
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  2. #2
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    Quote Originally Posted by thippli View Post
    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!)
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