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  "body": "\n## Statement\n\nLet $F$ be a [[free group|0KTW26C80J]].\nThen every [[subgroup|0X1JXT47GA]] $H$ of $F$ is a free group.\nMoreover, if $F$ has finite rank $n$ and $H$ has finite [[index|0DTXBQPA2S]] $e = [F : H]$, then $H$ is free of rank $1 + e(n - 1)$.\n\n## Remarks\n\nA topological proof: $F$ is the [[fundamental group|S1XK3HXHFX]] of a bouquet of circles, a graph.\nThen $H$ is the fundamental group of a [[covering space|3XZRRE165N]], which is again a graph.\nThe fundamental group of a connected graph is free.\nFor the rank formula, the covering graph of degree $e$ of a bouquet of $n$ circles has $e$ vertices and $en$ edges.\nSo its Euler characteristic is $e(1 - n)$.\nA connected graph with Euler characteristic $\\chi$ has free fundamental group of rank $1 - \\chi$.\nAn algebraic proof uses a Schreier transversal and gives explicit free generators, the Schreier generators.\nExample: in the free group $F(a, b)$, let $H$ be the kernel of the map to $\\Z/2$ sending $a$ and $b$ to $1$.\nIt has index $2$, so it is free of rank $1 + 2 \\cdot 1 = 3$; a free basis is $a^2$, $ab$ and $ab^{-1}$.\nSo a subgroup of a free group can have larger rank than the group.\nThe free group of rank $2$ contains free subgroups of every finite rank, and of countably infinite rank, such as its [[commutator subgroup|0ASGSTXSBW]].\nThe finite index hypothesis is needed in the rank formula: the commutator subgroup of $F(a, b)$ has infinite index and is not finitely generated.\n\n## Explanations\n\n### L1 human\n\nA free group is a group with no relations among its generators other than the obvious cancellations.\nThe theorem says that every subgroup of such a group is again free.\nThis is surprising, since subgroups of a free group can be complicated, but they never acquire genuine relations.\n\n### L2 human\n\nThe theorem is one of the starting points of combinatorial and geometric group theory.\nThe covering space proof is the model for [[Bass–Serre theory|6M731X66RD]], where groups acting on trees are studied, and the [[Kurosh subgroup theorem|RY8K5RQJSR]] is the analogue for subgroups of [[free products|TM7QS7P7X9]].\nThe rank formula is a form of the multiplicativity of the Euler characteristic under finite coverings.\n",
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