You are right and your friend is wrong.
For instance, look at the following sentences and at the explanations below to understand the difference between present perfect and past simple:
(a) Past simple: I lived in Florence for five years (... but I do not
live there anymore.)
(b) Present perfect: I have lived in Florence for five years (... and I
still live there now.)
(c) Past simple: I broke my glasses ... (but it does not matter. I
(d) Present perfect: I have broken my glasses ... (and so I can't see
You probably learned the difference between (a) and (b) years ago: one of the differences between past simple and past perfect is the 'time' of the verb, i.e. when it happened. The difference between (c) and (d) is harder to understand.
In (c) and (d), 'time', i.e. when the verb happened, is not really what separates the two sentences; it is possible that both (c) and (d) happened last month, this morning, or one second ago. What is important is that the event in (d) is considered more relevant to the situation now than the event in (c), which is why it is given in the present perfect.
That said, let us consider the first sentence "I sent you a letter a few days ago, I was wondering if you have received it." Here the person who asks the question would seem interested to talk about what he or she wrote in the letter. (Lett. 'd' in my example.)
In the sentence "I sent you a letter a few days ago, I was wondering if you received it," the person who asks the question wants to make sure himself or herself that the letter is being received. (Lett. 'c' in my example.)