gegeneinander
against each other
Expresses that two or more parties act as opponents in a competition, fight, or dispute.
Im Finale spielten die beiden besten Mannschaften gegeneinander.
In the final, the two best teams played against each other.

against each other
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against each other
Expresses that two or more parties act as opponents in a competition, fight, or dispute.
Im Finale spielten die beiden besten Mannschaften gegeneinander.
In the final, the two best teams played against each other.

The word is made up of 'gegen' (against) and 'einander' (each other). So it describes an action that two or more parties direct toward each other.
Die Kinder warfen den Ball gegeneinander.
The children threw the ball at each other.
Since 'gegeneinander' (against each other) expresses a reciprocal relationship, the subject of the sentence must always be plural or refer to a group.
Die beiden Teams treten morgen gegeneinander an.
The two teams will compete against each other tomorrow.
In the context of competition or dispute, 'gegeneinander' (against each other) is often used with verbs like 'spielen' (to play), 'kämpfen' (to fight), 'antreten' (to compete), or 'argumentieren' (to argue).
Im Debattierclub argumentieren die Mitglieder oft gegeneinander.
In the debate club, the members often argue against each other.
When referring to a collision, 'gegeneinander' (into each other) is used with verbs of movement. Typical examples are 'fahren' (to drive), 'laufen' (to run), 'prallen' (to crash), or 'stoßen' (to bump).
Auf der glatten Straße sind zwei Autos gegeneinander gerutscht.
On the slippery road, two cars slid into each other.
against / around
'gegen' and 'gegeneinander' can both express that people or things are not acting together, but as opponents, or that they collide. 'gegen' is a preposition and normally needs a noun or pronoun: 'gegen den Plan' (against the plan), 'gegen die Wand' (against the wall), 'gegen die andere Mannschaft' (against the other team). 'gegeneinander' is an adverb and describes a mutual situation between two or more sides: two teams play 'gegeneinander' (against each other), two cars collide 'gegeneinander' (into each other). Learners easily confuse the words because both have to do with conflict, competition, or impact. As a practical rule: if you want to say who or what the target or opponent is, use 'gegen'. If both sides are acting mutually as opponents, 'gegeneinander' often fits.

Im Finale sind die beiden besten Teams gegeneinander angetreten.
In the final, the two best teams competed against each other.
Unser Team spielt am Samstag gegen eine starke Mannschaft.
Our team is playing against a strong team on Saturday.