Left representation is isomorphic to group

Question

Let $G$ be a group and $G_L$ be its left representation, that is $G_L = \{g_L ; g_L(x)=gx\}$.

Show that $G$ is isomorphic to $G_L$.

Solution To show that $G$ is isomorphic to $G_L$ we need an isomorphism.

Let $$F: G \rightarrow G_L$$ be a map defined as $$g \mapsto g_L$$ Claim: F is an isomorphism.

Homomorphism: $$F(g+h) = (g+h)_L (x) = (g+h)x = gx+hx = F(g) + F(h)$$

1. Injective:
   Assume $F(g)= F(h) \implies g_L(x) =h_L(x) \forall x \in G \implies gx=hx \implies g=h $

2. Now I am stuck, can someone please help how to prove surjectivity?

Answer 1

Let $g_L\in G_L$. Then $g_L(x)=gx$, some $g\in G$ and all $x\in G$. But then $F(g)\color{red}{(x)}=g_L(x)$ for all $x\in G$. Hence $g_L=F(g)$.

I think your confusion is that you forgot the $\color{red}{(x)}$.