If we define $\mathbb{Z}\ltimes G$ to be $\mathbb{Z}\times G$, with multiplication as $$ (a,g)*(b,h)=(a+b,g^a\cdot h) $$ I want to show that $\mathbb{Z}\ltimes G$ is isomorphic to $\mathbb{Z}*G$(free product) by the relation $tg=gt$, where $t$ is the generator of $\mathbb{Z}$. I am thinking to define a map $(a,g)\mapsto(a,g)$ under the quotient. But I don’t think it will satisfy the multiplication.
I am really confused with the free products, sign. Thanks!