# Torsion-free group for a set of primes

From Groupprops

## Definition

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

No. | Shorthand | Explanation |
---|---|---|

1 | -torsion-free, for each prime | For any and any such that is the identity element of , must also be the identity element of . |

2 | -torsion-free for every -number | For any natural number and any such that is the identity element of , must also be the identity element of . |