Torsion-free group for a set of primes

Definition

Let $\pi$ be a set of primes. A group $G$ is termed $\pi$-torsion-free if it satisfies the following equivalent conditions:

No.	Shorthand	Explanation
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 $G$.