C_2 free-product C_2 is the infinite dihedral group, and the product of its generators has infinite order

Given: The objects and hypotheses in the example.

Suppose each $G_i$ has a presentation $\langle X_i\mid R_i\rangle$, with the alphabets replaced by disjoint copies. Then $\ast_iG_i\cong\left\langle\bigsqcup_iX_i\ \middle|\ \bigcup_iR_i\right\rangle.$ (A free product has the union presentation of presentations of its factors).

Every element of $\ast_{i\in I}G_i$ has a unique reduced syllable expression. The identity is represented by the empty word, and no nonempty reduced word represents the identity. (Normal form theorem for free products).

For every $n\in\mathbb N$, view $n$ as its canonical nonnegative integer and put $n\mathbb Z:=\{nk:k\in\mathbb Z\}$. Then the left cosets of $n\mathbb Z$ in $(\mathbb Z,+)$ are exactly the congruence classes modulo $n$, and coset addition is the published addition of congruence classes. Thus $(\mathbb Z,+)/n\mathbb Z=(\mathbb Z/n,+)$ as the same group on the same underlying set. This includes $n=0$ and $n=1$. (For every $n\in\mathbb N$, the congruence-class group $(\mathbb Z/n,+)$ is the quotient group $(\mathbb Z,+)/n\mathbb Z$).