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  "body": "\n## Definition\n\nLet $(G_i)_{i \\in I}$ be a family of [[groups|G5S1M9]], with $G_i \\cap G_j = \\emptyset$ for $i \\ne j$ after relabelling.\nA *reduced word* is a finite sequence $(g_1, \\ldots, g_m)$, with $m \\ge 0$, such that:\n\n1. each $g_k$ is a non-identity element of some $G_{i_k}$;\n2. consecutive letters lie in different groups: $i_k \\ne i_{k+1}$ for $1 \\le k \u003c m$.\n\nThe *free product* $\\mathop{\\ast}_{i \\in I} G_i$ is the set of reduced words.\nThe product of two words is their concatenation followed by reduction.\nTo reduce, replace two adjacent letters of the same $G_i$ by their product in $G_i$, and delete the letter if it is the identity.\nRepeat until the word is reduced.\nFor two groups it is written $G * H$.\n\n## Remarks\n\nThe product is well defined and associative, the empty word is the identity, and each $G_i$ embeds as the words of length at most $1$.\nEquivalently, $\\mathop{\\ast}_i G_i$ is the coproduct of the $G_i$ in the category of groups.\nThat is, for every group $K$ and [[homomorphisms|0JYTG5K9Y4]] $\\varphi_i \\colon G_i \\to K$, there is a unique homomorphism $\\mathop{\\ast}_i G_i \\to K$ restricting to each $\\varphi_i$.\nExample: $\\Z * \\Z$ is the free group of rank $2$.\nWith generators $a$ of the first factor and $b$ of the second, its reduced words are the reduced words in $a^{\\pm 1}, b^{\\pm 1}$.\nAnother example: $\\Z/2 * \\Z/3$ is isomorphic to the [[modular group|VEKV1EM77B]] $PSL_2(\\Z)$.\nNon-example: the direct product $\\Z \\times \\Z$ is not the free product $\\Z * \\Z$.\nIn $\\Z \\times \\Z$ the element $a b a^{-1} b^{-1}$ is the identity, while in $\\Z * \\Z$ it is a reduced word of length $4$.\nGiven presentations $G = \\langle S \\mid R \\rangle$ and $H = \\langle T \\mid Q \\rangle$ with $S \\cap T = \\emptyset$, one has $G * H = \\langle S \\cup T \\mid R \\cup Q \\rangle$.\n",
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