A free product has the union presentation of presentations of its factors

Given: The objects and hypotheses in the statement.

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 $⟨X\mid R⟩$ be a presentation, let $H$ be a group, and let $u:X\to H$ be a function. If the evaluation of every $r\in R$ under $u$ is $e_H$, then there is a unique homomorphism $\bar{u}:⟨X\mid R⟩⟶H$ with $\bar{u}([x])=u(x)$ for every $x\in X$. Moreover, $\bar{u}$ is surjective if and only if $u(X)$ generates $H$. (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

For a family $(G_i{)}_{i\in I}$, a free product is a group $F$ with homomorphisms ${\iota }_i:G_i\to F$ in the sense of def-group-homomorphism, such that for every group $H$ and every family of homomorphisms $f_i:G_i\to H$, there is a unique homomorphism $f:F\to H$ satisfying $f\circ {\iota }_i=f_i$ for all $i$. It is denoted ${\ast }_{i\in I}{G}_i$. Injectivity of the maps ${\iota }_i$ is not part of this definition. (The free product of an arbitrary family of groups).

Any two free products of the same family are connected by a unique isomorphism commuting with every canonical factor map. (Free products are unique up to a unique factor-compatible isomorphism).

In a presentation $⟨X\mid R⟩$ as in def-group-presentation, an element $r\in R\subseteq F(X)$ is called a defining relator. The equation $r=1$ that it imposes in the quotient is a defining relation. More generally, an equation $u=v$ may be recorded by the relator $u^{-1}v$. The published definition uses the common looser convention of calling the members of $R$ relations; both conventions define the same quotient group. A presentation is finitely generated when $X$ is finite, finitely related when $R$ is finite, and finite when both $X$ and $R$ are finite. A group is called finitely generated, finitely related, or finitely presented when it admits a presentation with the corresponding property. For finitely generated groups this agrees with generation by a finite subset in the sense of def-generated-subgroup. (Relators and relations; finitely generated, finitely related, and finite presentations).