Equivalence relation and subgroup I am taking abstract algebra now, and there's a lemma: Let $H$ be a subgroup of group $G$, for $a,b \in G$,define $a\sim b$ if $ab^{-1}\in H$, then it is an equivalence. I know how to prove it and how to use it in the prove of Lagrange theorem, but can anyone give me a more mathematical intuition explanation about it ? My book has an example,let $a,b$ belong the congruence class, then $a\sim b$ will be like $a-b \in H$ like $a-b\pmod n$, but it is a particular problem, can someone tell me a more general one? How does the lemma been created? 

The basic idea of that equivalence relation is that the equivalence classes of $\sim$ form what are called cosets. One way to talk about cosets is as equivalence classes of the relation you're talking about. The other way is to define, with $H \subset G$ a subgroup and $g \in G$, $gH = \{gh : h \in H\}$, i.e., the set of all products $gh$, with $g$ fixed and $h$ ranging over the subgroup $H$. It's worthwhile, and not too hard, to show that those two definitions of cosets are equivalent. 
As an example, let $G = \mathbb{Z}$, and $H = 10 \mathbb{Z}$. Then the coset $(1 + H) = \{1, 11, 21, 31, \dots$ } (note that we're using additive notation.) Hopefully that clears up some of your confusion. Assuming you're in an algebra class now, you'll be using cosets extensively, so you'll see their use soon.
