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  "body": "\n## Summary\n\nA torsion-free group is a [[group|G5S1M9]] in which the identity is the only element of finite [[order|0WE2NHZJ7B]].\nIt is the opposite extreme to a [[torsion group|JZNQTTMG58]].\nFor an abelian group $A$, the quotient of $A$ by its [[torsion subgroup|0DBWXQRXQJ]] is torsion-free.\nA finitely generated torsion-free abelian group is a [[free abelian group|0WJX8MGG37]] $\\Z^n$.\n\n## Explanations\n\n### L1 human\n\nIn a torsion-free group, no element other than the identity ever comes back to the identity when you repeat it.\nThe integers are the basic example: starting from $3$ and adding $3$ again and again, you get $6, 9, 12, \\dots$ and never $0$.\nA quarter turn of a circle comes back to the start after four steps.\nSo a group that contains a quarter turn is not torsion-free.\nTorsion-free groups are the groups with no hidden clocks inside.\n",
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