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  "body": "\n## Statement\n\nFor $n \\ge 1$, let $\\Sigma_n$ be the [[symmetric group|0RJMNEN3BJ]] on $n$ letters, and let $\\Sigma_\\infty = \\bigcup_n \\Sigma_n$ be the infinite symmetric group, with $B\\Sigma_\\infty$ its [[classifying space|7XRBR7AV3C]].\nLet $QS^0 = \\operatorname{colim}_n \\Omega^n S^n$, an [[infinite loop space|A7VB2NSGJ2]], and let $Q_0 S^0$ be its path component of the basepoint.\nThen there is a natural map $B\\Sigma_\\infty \\to Q_0 S^0$ that induces an isomorphism\n$$H_*(B\\Sigma_\\infty; \\Z) \\cong H_*(Q_0 S^0; \\Z)$$\nin [[singular homology|M0Q5CFJ7ZH]].\nEquivalently, $QS^0$ is the group completion of the topological monoid $\\coprod_{n \\ge 0} B\\Sigma_n$, and $\\Z \\times B\\Sigma_\\infty^+ \\simeq QS^0$, where $B\\Sigma_\\infty^+$ is Quillen's plus construction.\n\n## Remarks\n\nBarratt and Priddy, and independently Quillen, proved the theorem in the early 1970s.\nOne proof uses the group completion theorem: the homology of the group completion of $M = \\coprod_n B\\Sigma_n$ is $H_*(M)[\\pi_0(M)^{-1}] = \\Z[\\Z] \\otimes H_*(B\\Sigma_\\infty)$.\nThe other input is that $\\coprod_n B\\Sigma_n$ is the free $E_\\infty$ space on a point, and that the group completion of the free $E_\\infty$ space on $S^0$ is $QS^0$.\n\nAt work: taking homotopy groups gives $\\pi_k(B\\Sigma_\\infty^+) \\cong \\pi_k^s$ for $k \\ge 1$, the [[stable homotopy groups of spheres|HGZAYPQBXW]].\nFor $k = 1$ this says $\\pi_1^s \\cong \\Z/2$, which matches the abelianization $\\Sigma_\\infty / A_\\infty \\cong \\Z/2$, where $A_\\infty$ is the infinite alternating group and the quotient map is the sign of a permutation.\n\n## Explanations\n\n### L1 human\n\nThe theorem links two very different objects: the symmetric groups, which permute finite sets, and the stable homotopy groups of spheres.\nIt says that, after the plus construction, the classifying space of the infinite symmetric group has exactly the stable homotopy groups of spheres as its homotopy groups.\nFor this reason the stable homotopy of spheres is often described as the algebraic K-theory of the \"field with one element\", modelled by finite sets.\n\n### L2 human\n\nSince $\\pi_1(B\\Sigma_\\infty^+)$ is the abelianization of $\\Sigma_\\infty$, the theorem recovers $\\pi_1^s \\cong \\Z/2$, and the higher groups give a description of all stable stems in terms of permutations.\nTogether with Nakaoka's homological stability for symmetric groups, it computes $H_*(Q_0 S^0)$ from the homology of the groups $\\Sigma_n$.\nQuillen's analogue for general linear groups over finite fields led to his computation of the algebraic K-theory of finite fields.\n",
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