Again, let be a group and a subgroup of . Then we can define a binary relation on elements of , called , as follows:
if and only if there is some such that .
That is, for any two elements , either , or not: yes, if you can get from to by combining (on the right) with some element in , and otherwise, no. Note that given any two elements , it is always possible to get from to by combining with some element of : in particular, . But this might not be an element of .
Now, an equivalence relation on a set is a relation with the following three properties:
- Reflexivity: every is related to itself, that is, .
- Symmetry: If , then also .
- Transitivity: If and , then .
The usual equality relation satisfies these properties: things are always equal to themselves; if then ; and if and then . The notion of an equivalence relation is a way to talk about more general kinds of equality.
Let’s prove that is an equivalence relation. This is really cool because it turns out that the three properties of an equivalence relation each follow from one of the three properties of a group!
- Reflexivity. We have to show that any is related to itself, that is, . By definition this means there is some such that . Well, that’s easy: since is a group, it has to contain the identity element , and .
- Symmetry. Suppose , that is, for some . Then we have to show , that is, there is some (which could be different from ) such that . Well, since is a group, it has to contain inverses. We can combine both sides of with to obtain —so the we are looking for is precisely .
- Transitivity. Suppose and . That means there are for which and . We want to show that , that is, for some . Substituting for , we find that (note how we used the third property of a group, associativity). So the we are looking for is just , which has to be in since is closed under the binary operation.
So for a given subgroup , this relation defines a sort of “equality with respect to ” on the elements of (whatever that means!). As for an example—consider again the subgroup . Which elements of are related to each other under ? What do you notice?