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 $⟨X_i\mid R_i⟩$, with the alphabets replaced by disjoint copies. Then ${\ast }_i{G}_i\cong ⟨{⨆}_i{X}_i|{\bigcup }_i{R}_i⟩\mathrm{.}$ (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 $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}=(\mathbb{Z}\mathrm{/}n,+)$ 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 $(\mathbb{Z}\mathrm{/}n,+)$ is the quotient group $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}$).