A free product with amalgamation has the factor presentations plus the amalgamating relations

Given: The objects and hypotheses in the statement.

If $f:K\to G$ and $h:K\to H$ are injective homomorphisms, their pushout is called the free product with amalgamation and is denoted $G{\ast }_{K}H$. The quotient construction is thm-group-pushout-as-an-amalgamated-quotient, and injectivity means the trivial-kernel condition of thm-group-homomorphism-injective-iff-trivial-kernel. The notation anticipates identifying $K$ with its two images, but injectivity of the canonical maps $G,H\to G{\ast }_{K}H$ is a theorem, not part of this definition. (Free products with amalgamation along monomorphisms).

For homomorphisms $f:K\to G$ and $h:K\to H$, let $N$ be the normal closure in $G\ast H$ of $\{j_G(f(k))j_H(h(k){)}^{-1}:k\in K\}\mathrm{.}$ Then $(G\ast H)\mathrm{/}N$, with the induced factor maps $j_G$ and $j_H$, is a pushout of $f$ and $h$. (A group pushout is the quotient of a free product by the amalgamating relations).

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).

Let $F(X)$ be a free group and let $R\subseteq F(X)$ be a set of words, called relations. The group with presentation $⟨X\mid R⟩:=F(X)\mathrm{/}⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}{⟩}_{F(X)}$ is the quotient by the normal closure of $R$. The members of $X$ are its generators. In this quotient, every relation in $R$ becomes the identity, as do all consequences forced by normality. (Group presentation by generators and relations).

Let $G$ be a group and $R\subseteq G$. Then $⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}{⟩}_G=\{g_1{r}_1^{{\epsilon }_1}{g}_1^{-1}\cdots g_n{r}_n^{{\epsilon }_n}{g}_n^{-1}:n\in \mathbb{N},g_i\in G,r_i\in R,{\epsilon }_i\in \{1,-1\}\}\mathrm{.}$ For $n=0$ the displayed product is the identity. Replacing every conjugator $g_i$ by $g_i^{-1}$ gives the equivalent convention $g_i^{-1}{r}_i^{{\epsilon }_i}{g}_i$. (The normal closure of $R$ is the set of finite products of conjugates of elements of $R$ and their inverses).