Proof of left adjoint for Group to Set

Question

I'm currently trying to prove that the forget functor from the Group of groups to Set has a left adjoint. Doing some reading I am sure I need to use the concept of a Free Group but coming from a Comp Sci background I'm struggling to form my proof. I was wondering if anyone could point me in the direction of a complete proof for this to assist me in understanding the concept?

Thanks a lot.

Answer 1

Yes, you are right, free groups ftw.

So consider a set $X$ and take a free group $\mathbb{F}(X)$ generated by the set $X$.

Now first of all $\mathbb{F}$ is functorial. If $f:X\to Y$ is a (set) function then

$$\mathbb{F}(f):\mathbb{F}(X)\to \mathbb{F}(Y)$$ $$\mathbb{F}(f)(x_1\cdots x_n)=f(x_1)\cdots f(x_n)$$

is a group homomorphism.

The left adjointness of $\mathbb{F}$ is given by the fact that any (set) function $f:X\to G$ where on the right side we have a group (although we treat it as a set, i.e. via forgetful functor) can be extended to a group homomorphism

$$F:\mathbb{F}(X)\to G$$ $$F(x_1\cdots x_n)=f(x_1)\cdots f(x_n)$$