The kernels of the amalgamating maps are killed in the opposite canonical maps to a group pushout

Given: The objects and hypotheses in the statement.

Given homomorphisms $f:K\to G$ and $h:K\to H$ as in def-group-homomorphism, a pushout is a group $P$ with homomorphisms $i_G:G\to P$ and $i_H:H\to P$ such that $i_G\circ f=i_H\circ h$, and such that every compatible pair $u:G\to Q$, $v:H\to Q$ factors through a unique $w:P\to Q$ with $w\circ i_G=u$ and $w\circ i_H=v$. The maps $f,h$ need not be injective. (Pushouts of group homomorphisms).

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).

Let $G$ and $G^{\mathrm{\prime }}$ be groups with identities $e$ and $e^{\mathrm{\prime }}$, and let $f:G\to G^{\mathrm{\prime }}$ be a group homomorphism (def-group-homomorphism), so $f(xy)=f(x)f(y)$ for all $x,y\in G$. Then: 1. $f(e)=e^{\mathrm{\prime }}$; 2. $f(g^{-1})=f(g{)}^{-1}$ for every $g\in G$; 3. $f(g^n)=f(g{)}^n$ for every $g\in G$ and every $n\in \mathbb{Z}$, powers being those of def-group-power. For monoid homomorphisms the analogue of claim 1 is false, so preservation of the identity has to be part of the definition: the map $u:\mathbb{Z}\to \mathbb{Z}$ with $u(x)=0$ for every $x$ satisfies $u(xy)=u(x)u(y)$ for the multiplicative monoid $(\mathbb{Z},\cdot ,1)$, yet $u(1)=0\mathrm{\ne }1$. (A group homomorphism automatically satisfies $f(e)=e^{\mathrm{\prime }}$ and $f(g^{-1})=f(g{)}^{-1}$, and $f(g^n)=f(g{)}^n$ for every $n\in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed).