Does every group have a representation?

Question

For any set $A$, we can give it an group structure and make it a free group. For example: $$\mathbb Z=<a;aa^{-1}=1>$$

Further more, we can introduce some relation on it: $$\mathbb Z_2=<a;a^2=1>$$ $$\mathbb Z_2\times{\mathbb Z_3}=<a,b;a^2=b^3=1,ab=ba>$$

So if a group can be represented by the form: $$<a_1,a_2...;r_1,r_2...>$$

then we call that it has a representation.

Question: Does every group have a representation?

Below is my attempt:

Let $G$ be a group. If we remove the group structure of $G$, then we can give it a trivial free relation: $G^{*}=<g;gg^{-1}=1>$.
Then we consider the map $$\sigma:G^{*}\rightarrow{G}$$ $$g_1^{\varepsilon_1}...g_n^{\varepsilon_n}\mapsto{g_1^{\varepsilon_1}...g_n^{\varepsilon_n}}$$

So now I'm confused. Is the map well-defined? And what should I do next step?

Answer 1

Yes, let $X:=\{x_g\}_{g\in G}$ be the set of variables and take relations $x_gx_h=x_u$ whenever $gh=u$.

If you wish to generate it as semigroup you indeed need the extra symbols $x_g^{-1}$ and relations $x_gx_g^{-1}=1$.