Les lions se sont battus entre eux pour obtenir de la nourriture.

English Translation

The lions fought each other to get food.

parmi should also be an acceptable answer

There is a subtle nuance though, which makes parmi not work so well in the original sentence.

Se battre entre eux = all involved.

Parmi is more restrictive. It suggests that you are talking about a subset of the group.

A similar phrase properly using it would be:

Parmi eux, seuls les plus jeunes lions se battent.

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I thought entre was for 2 and parmi was used for more than two, like between and among for English. Thanks for the clarification!