Learn Hardcore German: In unserer Lerngruppe im Sprachkurs vergleichen wir unseren Wortschatz und helfen einander. - In our study group in the language course we compare our vocabulary and help each other.

Dative = location (where?): in unserer Lerngruppe – in our study group (we are already there)
Accusative = direction (where to?): in unsere Lerngruppe – into our study group (movement towards it)

Here we’re talking about what we do within the group, so it’s a static location, and therefore dative:

die Lerngruppe (nom.) → der Lerngruppe (dat.)
with unser: unserer Lerngruppe (feminine dative singular)

Why does the verb come before „wir“ in „vergleichen wir unseren Wortschatz“?

German main clauses follow the verb-second rule: the conjugated verb is always in the second position. The “first position” can be:

the subject: Wir vergleichen unseren Wortschatz.
or any other element, like a time or place phrase:
In unserer Lerngruppe im Sprachkurs vergleichen wir unseren Wortschatz.

Here, „In unserer Lerngruppe im Sprachkurs“ is position 1 (as one big chunk), so the verb „vergleichen“ must be position 2, and the subject „wir“ moves after the verb.

What exactly is a „Lerngruppe“, and how is it different from just „Gruppe“ or „Kurs“?

Lerngruppe = a study group: a small group of people who meet specifically to learn/study together.
Gruppe = any group; it doesn’t necessarily have a learning purpose.
Kurs = the course/class itself (organized by a school, institution, etc.).

So:

im Sprachkurs – in the language course
in unserer Lerngruppe im Sprachkurs – in our study group within that course

What does „im Sprachkurs“ mean exactly, and what is this „im“?

„im“ is the contraction of „in dem“.

Sprachkurs is masculine: der Sprachkurs
Dative singular masculine: dem Sprachkurs
in dem Sprachkurs → im Sprachkurs

Meaning: „in the language course“.

It again uses dative, because it’s about being located in the course, not moving into it.

Could I say „beim Sprachkurs“ instead of „im Sprachkurs“? Is there a difference?

Both are possible, but there’s a nuance:

im Sprachkurs (in dem Sprachkurs)
Focus on being inside the course as a setting or part of it.
beim Sprachkurs (bei dem Sprachkurs)
Focus on being at/around the event of the course, more like “during the course” or “at the time/place of the course”.

In your sentence, „im Sprachkurs“ fits very naturally, because the study group is a part of that course.

Why is it „unseren Wortschatz“ and not just „unser Wortschatz“?

Because „unseren Wortschatz“ is a direct object in the accusative case.

Wortschatz is masculine: der Wortschatz
Accusative singular masculine: den Wortschatz
With unser: unseren Wortschatz

Pattern:

Nominative: Unser Wortschatz ist groß. – Our vocabulary is big.
Accusative: Wir vergleichen unseren Wortschatz. – We compare our vocabulary.

Why is „Wortschatz“ singular? In English we might say “our vocabularies”.

In German, „Wortschatz“ is very often used in the singular even if you’re talking about several people.

Wir vergleichen unseren Wortschatz.
= We compare (the level/extent of) our vocabulary.

„Wortschätze“ (plural) exists, but is used when you really mean distinct sets of vocabulary, e.g.:

Die Wortschätze von Kindern und Erwachsenen sind sehr unterschiedlich.
The vocabularies of children and adults are very different.

In everyday speech about language learning, the singular is far more natural.

Is there any difference between „Wortschatz“ and „Vokabular“?

They’re close in meaning and often interchangeable:

Wortschatz – the more common, everyday German word for vocabulary (both active and passive).
Vokabular – a bit more “technical” or foreign-sounding; used in linguistic contexts, or for style-specific vocabulary (juristisches Vokabular = legal vocabulary).

In your sentence, „Wortschatz“ is the most idiomatic choice.

Why do we say „helfen einander“ instead of „helfen uns“? Are both correct?

Both are correct, but there’s a nuance:

wir helfen einander – we help each other
Uses the reciprocal pronoun „einander“ (= each other).
wir helfen uns – literally we help us, but usually understood as we help each other from context.

Differences:

„einander“ makes the mutual nature of the action very explicit.
„uns“ is more neutral and depends on context; it could also be interpreted as “we help ourselves” (reflexive).

In a sentence about a group learning together, both are fine, but „einander“ sounds nicely clear and a bit more formal/precise.

I learned that „helfen“ takes the dative. Why doesn’t „einander“ look dative here?

You’re right that „helfen“ normally takes a dative object:

Ich helfe dir. – I help you. (dative)
Wir helfen den anderen. – We help the others. (dative)

With „einander“, German uses a fixed, invariable form. It does not change its ending for case, but it still functions as a dative (or accusative, depending on the verb).

So in:

Wir helfen einander.

„einander“ is dative in function, but not in form. German simply doesn’t mark the case on this word; the verb „helfen“ and the sentence structure tell you it’s dative.

Do I need a comma before „und helfen einander“?

No comma is needed in the original version:

In unserer Lerngruppe im Sprachkurs vergleichen wir unseren Wortschatz und helfen einander.

Here, „vergleichen“ and „helfen“ share the same subject (wir) and form a single clause with two verbs joined by „und“ → no comma.

If you add a new subject explicitly, then you need a comma:

In unserer Lerngruppe im Sprachkurs vergleichen wir unseren Wortschatz, und wir helfen einander.

Now there are two main clauses, so you must use a comma.

Can I move „im Sprachkurs“ somewhere else in the sentence?

Yes, several positions are possible and grammatical; the meaning stays almost the same, only the emphasis shifts slightly. For example:

Im Sprachkurs in unserer Lerngruppe vergleichen wir unseren Wortschatz und helfen einander.
In unserer Lerngruppe vergleichen wir im Sprachkurs unseren Wortschatz und helfen einander.
Wir vergleichen in unserer Lerngruppe im Sprachkurs unseren Wortschatz und helfen einander.

All are acceptable. The original version:

In unserer Lerngruppe im Sprachkurs vergleichen wir unseren Wortschatz und helfen einander.

nicely presents the place setting first (study group inside a course), then what you do there.