Free product

Remarks

The product is well defined and associative, the empty word is the identity, and each $G_i$ embeds as the words of length at most $1$. Equivalently, $\mathop{\ast}_i G_i$ is the coproduct of the $G_i$ in the category of groups. That is, for every group $K$ and homomorphisms $\varphi_i \colon G_i \to K$, there is a unique homomorphism $\mathop{\ast}_i G_i \to K$ restricting to each $\varphi_i$. Example: $\Z * \Z$ is the free group of rank $2$. With generators $a$ of the first factor and $b$ of the second, its reduced words are the reduced words in $a^{\pm 1}, b^{\pm 1}$. Another example: $\Z/2 * \Z/3$ is isomorphic to the modular group $PSL_2(\Z)$. Non-example: the direct product $\Z \times \Z$ is not the free product $\Z * \Z$. In $\Z \times \Z$ the element $a b a^{-1} b^{-1}$ is the identity, while in $\Z * \Z$ it is a reduced word of length $4$. Given presentations $G = \langle S \mid R \rangle$ and $H = \langle T \mid Q \rangle$ with $S \cap T = \emptyset$, one has $G * H = \langle S \cup T \mid R \cup Q \rangle$.