This is pretty straight forward, just check the three properties of an equivalence relation. Recall an equivalence relation is reflexive, symmetric, and transitive. Here's a start.

Let

If , then since is a subgroup, (symmetry)

Now you need only show:

(reflexivity)

and

If and , then (transitivity)

For the second part, show containment both ways, and . It can be done in one fell swoop with a short string of equivalences if implications in both directions seems redundant.