Мы уже купили по одному билету на концерт.

It means one ticket each or one ticket apiece.

So Мы уже купили по одному билету на концерт means that every person in мы bought one ticket. It does not mean that the group bought only one single ticket total.

Because по одному is a very common Russian way to express distribution: one each.

Compare:

• Мы купили один билет = We bought one ticket (just one total)
• Мы купили по одному билету = We bought one ticket each

The preposition по often gives this distributive meaning.

Because after по in this distributive meaning, Russian commonly uses the dative case.

So:

• билет = nominative
• билету = dative singular

That is why you get:

• по одному билету

Both одному and билету are in the dative singular.

For the same reason: it has to agree with билету in case, gender, and number.

• билет is masculine singular
• after по, we use the dative
• so один becomes одному

That gives:

• по одному билету

Because the phrase is built around the idea one each, and Russian uses the noun in the singular in this pattern.

So even though the total number of tickets is more than one, the sentence treats it as a repeated unit:

• one ticket for one person
• one ticket for another person
• and so on

That is why Russian says по одному билету, not по одним билетам.

Купили is the perfective past tense of купить, and it shows a completed action: the tickets have already been bought.

• купили = bought / have bought
• покупали would usually suggest an ongoing, repeated, or background action, depending on context

Here, the point is the result: the purchase is done. So купили is the natural choice.

Уже means already.

It tells you that the action happened before now, often with the sense that this may be earlier than expected or that the matter is settled.

So:

• Мы купили по одному билету на концерт = We bought one ticket each for the concert
• Мы уже купили по одному билету на концерт = We’ve already bought one ticket each for the concert

Because in Russian, with events like concerts, shows, lectures, and exhibitions, the usual preposition is на.

So Russian says:

• билет на концерт = a ticket to/for the concert
• пойти на концерт = to go to the concert

Using в here would sound wrong.

Yes, купили already tells you the subject is we or they, depending on context, because it is past plural.

But Russian often still includes the subject pronoun for clarity, contrast, or normal conversational style.

So:

• Купили по одному билету на концерт can work if the subject is obvious from context
• Мы уже купили по одному билету на концерт is clearer and more neutral on its own

Yes. Russian word order is flexible, and changing it can shift emphasis.

For example:

• Мы уже купили по одному билету на концерт.
  Neutral, natural order.

• Мы по одному билету на концерт уже купили.
  More marked; may emphasize the fact that the tickets are already bought.

• По одному билету на концерт мы уже купили.
  Focuses more on one ticket each.

The original sentence is a very natural neutral version.

No. The phrase по одному билету specifically means one each.

If you wanted to say the group bought only one ticket total, you would say:

• Мы купили один билет на концерт.

So the по makes an important difference.

Yes. For example:

• Каждый из нас уже купил билет на концерт. = Each of us has already bought a ticket for the concert.
• Мы уже купили билеты на концерт. = We have already bought tickets for the concert.

But these are not exactly the same:

• по одному билету clearly emphasizes one each
• билеты just says tickets, without stressing the exact distribution
• каждый из нас sounds slightly more explicit and individual

In most contexts, the best natural English translation is We’ve already bought one ticket each for the concert.

Russian past tense does not directly distinguish between English simple past and present perfect, so купили can match either, depending on context.

Because of уже and the completed-result meaning, English often prefers have already bought.