free groups question

Question

I need help to solve this question

If $f:G_1\to G_2$ and $g:H_1\to H_2$ are homomorphisms of groups, then there is a unique homomorphism $h:G_1*H_1\to G_2*H_2$ such that $h_{G_1}=f$ and $h_{H_1}=g$

I don't have much experience solving free groups questions, I need a hint or something.

Thanks a lot

Answer 1

It's about free product of groups. The free product of $G_1$ and $H_1$ is the freeest 'formal' group that disjointly contains $G_1$ and $H_1$. Its elements are then written in the form $x_1y_1x_2y_2\dots$ for $x_i\in G_1,\,y_i\in H_1$ (it might start with a $H_1$-element as well, but that might also be considered with $x_1=e$ (the unit of $G_1$).

Let the mapping $h$ act on $G_1$ as $f$ and on $H_1$ as $g$. That is, for a generic element $x_1y_1x_2y_2\dots$ of $G_1*H_1$, define $$h(x_1y_1x_2y_2\dots):=f(x_1)g(y_1)f(x_2)g(y_2)\dots$$