Let \((G_\alpha)_{\alpha\in A}\) be a family of groups. A reduced word is a finite sequence \(g_1g_2\cdots g_k\), \(k\geq 0\), in which each letter \(g_i\) lies in \(G_{\alpha_i}\sm\ts{1}\) for some \(\alpha_i\in A\) and \(\alpha_i\neq\alpha_{i+1}\) for \(1\leq i<k\). The free product \(\ast_{\alpha\in A} G_\alpha\) is the set of reduced words, with product given by concatenating two words and then reducing: adjacent letters from the same \(G_\alpha\) are replaced by their product in \(G_\alpha\), and letters equal to \(1\) are deleted, until the word is reduced.
The free product \(\ast_\alpha G_\alpha\) is a group whose identity is the empty word, and each \(G_\beta\) is a subgroup via one-letter words. For every group \(H\) and homomorphisms \(\varphi_\alpha\colon G_\alpha\to H\) there is a unique homomorphism \(\varphi\colon\ast_\alpha G_\alpha\to H\) restricting to \(\varphi_\alpha\) on each \(G_\alpha\), so \(\ast_\alpha G_\alpha\) is the coproduct of the \(G_\alpha\) in \(\Grp\).
For a set \(S\), the free group on \(S\) is isomorphic to \(\ast_{s\in S}\ZZ\).
See [@Hat02].