Originally Posted by

**NonCommAlg** that's not correct! on every set the relation, as you defined, is an equivalence relation. so i'm sure the relation that you're talking about is already defined somewhere in your book

but you missed it. you need to find out how exactly the relation is defined, i.e. g ~ h if and only if what? for example you may define g ~ h iff gk = kh, for some k in G. then, with this

definition, ~ becomes an equivalence relation in G and if g ~ h for all g and h in G, then G has to be trivial because g ~ 1 if and only if gk = k, for some k in G. since G is a group, we'll

get that g = 1. so G has only one element, i.e. the identity element 1.