You don't. You have to define a specific f and show that that f is both injective and surjective. If you simply assert the existance of such an f, you are assuming what you want to prove.
Since how about trying ?
Let G be a group and let be a subgroup. If , then is defined as
where .
Show that the sets and have the same cardinality.
Hint: sets and have the same cardinality if and only if there exists an injective and surjective map .
Attempt: So, in order to show that the mapping f is injective I have to show that
(for and ).
But how can I deduce that when I don't know what the function f is?