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 $G$.