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.