Herstein ask us to find a group G, H<G and g in G such that gHg* is in H but gHg*/=H.
I can't imagine that this is true (of course if Herstein says it is true then he is correct) because of the following argument.
o(gH)=o(H) since gH has no duplications because G has the cancellation property. Now gH has o(H) distinct elements of G. Then o(gHg*)=o(H) again because of the cancellation law in G. Does it not follower that gHg* = H??? Where am I going wrong in my argument?
Thanks for any help offered,