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?