What is the connection between equivalence relations and groups? I have taken a group theory course, and we initially learned about equivalence relations. But I do not see the connection between equivalence relations and groups in group theory.  Can someone please explain to me why they are related?
 A: You know that the set of non-zero real  numbers form a group under multiplication.
You also know from high school the following rules about sign, for multiplication :
$$+\times + = +;\quad +\times -= -\times+= -;\quad -\times - = +  $$
One can restate this as $\{-, +\}$ forms a group (of order 2) with the above rules for multiplication.  (with $+$ as the identity element).
Go back to the original group of non-zero numbers, and consider the equivalence relation that puts all positive numbers in one class and the negative numbers in another class; give names to these classes  (symbolically) as $+$ and $-$. Now multiplication of real numbers leads to multiplication of the equivalence classes. The rules are already stated.
One looks for equivalence classes in groups such that  group operation on them leads consistently to group operation of the equivalence classes. 
The equivalence classes given by cosets of a normal subgroups are precisely such ones.
A: Two application of equivalence relations to group theory:
1) Being in the same left-sided coset of a subgroup is an equivalence relation.
2)  Belonging to the same orbit under a group action is an equivalence relation.  If the group acts on itself by conjugation, the equivalence classes are called conjugacy classes and play a role in linear representations of groups (see Serre's Linear representations of finite groups).
