# Torsion-free group for a set of primes

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Let $\pi$ be a set of primes. A group $G$ is termed $\pi$-torsion-free if it satisfies the following equivalent conditions:
1 $p$-torsion-free, for each prime $p \in \pi$ For any $p \in \pi$ and any $x \in G$ such that $x^p$ is the identity element of $G$, $x$ must also be the identity element of $G$.
2 $n$-torsion-free for every $\pi$-number $n$ For any natural number $n$ and any $x \in G$ such that $x^n$ is the identity element of $G$, $x$ must also be the identity element of $G$.