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  "body": "\n## Summary\n\nThe free product $G * H$ of two [[groups|G5S1M9]] is the group generated by $G$ and $H$ with no relations between them.\nIts elements are reduced words that alternate between non-identity elements of $G$ and of $H$.\nIt is the coproduct in the category of groups.\nThe free group of rank $n$ is the free product of $n$ copies of $\\Z$.\n\n## Explanations\n\n### L1 human\n\nTo form a free product, put two groups side by side and let their elements multiply freely, without imposing any commutation between them.\nAn element is a word such as $g_1 h_1 g_2 h_2$, with letters taken alternately from the two groups.\nFor example, $\\Z/2 * \\Z/2$ is generated by two reflections $s$ and $t$, and the words $st$, $stst$, $ststst$ are all different, so the group is infinite.\nIn topology, the fundamental group of two well-behaved spaces, such as graphs, glued at a point is the free product of their fundamental groups.\n",
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