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  "body": "\n## Definition\n\nLet $A$ be an [[abelian group|082GHNPJ53]].\nThe torsion subgroup of $A$ is $A_T = \\{a \\in A : a \\text{ has finite order}\\}$.\nThe group $A$ is a torsion group when $A_T = A$, and it is torsion-free when $A_T = \\{e\\}$.\n\n## Remarks\n\nThe set $A_T$ is a [[subgroup|0X1JXT47GA]] because $A$ is commutative: if $a^m = e$ and $b^n = e$, then $(a b)^{m n} = e$.\nIn a nonabelian group, the elements of finite order need not form a subgroup. In $\\langle x, y \\mid x^2, y^2 \\rangle$, the product $x y$ has infinite order.\nThe [[quotient group|0KYX2N69J0]] $A / A_T$ is torsion-free, and $A_T$ is a characteristic subgroup of $A$.\nFor $A = \\mathbb{Q}/\\mathbb{Z}$, each element has finite order, so $A_T = A$, but the orders have no upper bound.\n\n## Explanations\n\n### L1 human\n\nThe torsion subgroup collects the elements that return to the identity after finitely many steps.\nIn an abelian group, these elements form a subgroup.\nRemoving them leaves a group in which only the identity has finite order.\n",
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