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  "body": "\n## Statement\n\nLet $A$ and $B$ be groups, and let $G = A * B$ be their [[free product|TM7QS7P7X9]].\nFor a [[finitely generated group|082DWWXMRJ]] $X$, let $\\rank X$ be the least number of elements that generate $X$.\n\n1. Let $F$ be a [[free group|0KTW26C80J]] of finite rank, and let $\\varphi \\colon F \\to G$ be a [[surjective|01RVCMSN5N]] [[homomorphism|0JYTG5K9Y4]]. Then $F = F_1 * F_2$ for some subgroups $F_1$ and $F_2$ of $F$ with $\\varphi(F_1) = A$ and $\\varphi(F_2) = B$.\n2. If $A$ and $B$ are finitely generated, then $\\rank(A * B) = \\rank A + \\rank B$.\n\n## Remarks\n\nPart 2 follows from part 1.\nIf $G$ is generated by $n$ elements, take $F$ free of rank $n$ and $\\varphi$ onto $G$.\nThen $F_1$ and $F_2$ are free factors of $F$, so their ranks add up to $n$, and they map onto $A$ and $B$, so $\\rank A + \\rank B \\le n$.\nThe other inequality is clear, since generators of $A$ and of $B$ together generate $A * B$.\nProofs of part 1 use Stallings foldings or [[Bass–Serre theory|6M731X66RD]].\n\nExample: $\\Z/2 * \\Z/3$ has rank $2$, and $\\Z^2 * \\Z$ has rank $3$.\n\nThe free product cannot be replaced by the direct product.\nThe groups $\\Z/2$ and $\\Z/3$ have rank $1$ each, but $\\Z/2 \\times \\Z/3 \\cong \\Z/6$ is cyclic, so its rank is $1$, not $2$.\n\n## Explanations\n\n### L1 human\n\nThe rank of a group is the smallest number of elements that generate it.\nThe theorem says that for a free product, the ranks just add.\nNo clever choice of generators can save anything when two groups are glued with no relations between them.\n\n### L2 human\n\nThe first part says that every way of mapping a free group onto $A * B$ splits along a free product decomposition of the free group.\nThe rank formula is the usual form of the theorem.\nIt implies that every nontrivial finitely generated group is a free product of finitely many freely indecomposable groups.\nHere a group is freely indecomposable if it is nontrivial and not a free product of two nontrivial groups.\nThe decomposition stops because each split lowers the ranks of the factors.\nIt fits with the [[Kurosh subgroup theorem|RY8K5RQJSR]] as one of the basic facts about free products.\n",
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