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  "body": "\n## Statement\n\nLet $(A_i)_{i \\in I}$ be a family of groups, and let $G = \\mathop{*}_{i \\in I} A_i$ be their [[free product|TM7QS7P7X9]].\nLet $H$ be a [[subgroup|0X1JXT47GA]] of $G$.\nThen there are a [[free group|0KTW26C80J]] $F$, groups $B_j$ for $j \\in J$, and elements $g_j \\in G$ with the following property.\nEach $B_j$ is a subgroup of some factor $A_{i(j)}$, and\n$$H = F * \\mathop{*}_{j \\in J} g_j B_j g_j^{-1}.$$\nIn words, $H$ is the free product of a free group and of [[conjugates|0DXD4QSZ8A]] of subgroups of the factors.\n\n## Remarks\n\nOne proof uses [[Bass–Serre theory|6M731X66RD]].\nThe free product $G$ acts on a tree with trivial edge stabilisers, and the vertex stabilisers are the conjugates of the factors $A_i$.\nThe subgroup $H$ acts on the same tree, so it is the fundamental group of a [[graph of groups|DC1CQMD3WK]] with trivial edge groups.\nSuch a group is the free product of its vertex groups and of a free group, the fundamental group of the underlying graph.\nA topological proof uses covering spaces of a wedge of spaces whose fundamental groups are the $A_i$.\n\nExample: if every $A_i$ is infinite cyclic, then $G$ is free.\nEach subgroup of $\\Z$ is trivial or infinite cyclic, so each $g_j B_j g_j^{-1}$ is free, and $H$ is free.\nThis recovers the [[Nielsen–Schreier theorem|W97X7WX5G8]].\nAnother example: in $\\Z/2 * \\Z/3$, which is isomorphic to $PSL(2, \\Z)$, every subgroup without elements of finite order is free.\n\n## Explanations\n\n### L1 human\n\nA free product glues groups together with no relations between them.\nThe theorem says that every subgroup of such a product is again built in the same simple way.\nIt is a free product of pieces taken from the original factors, moved around by conjugation, together with a free group.\n\n### L2 human\n\nThe theorem is the main structure result for subgroups of free products.\nIt generalises the Nielsen–Schreier theorem, which says that subgroups of free groups are free.\nIt implies that a subgroup that meets every conjugate of every factor only in the identity is free.\nIn Bass–Serre theory it is the special case of trees with trivial edge stabilisers; the general case describes subgroups of amalgamated products and [[HNN extensions|YEYERRY37K]].\n",
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