Torsion-free group for a set of primes

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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.