Definition of left regular representation of $G$. I don't really understand the definition of left regular representation of $G$. Could anyone give me some explanation or examples?

The permutation representation afforded by left multiplication on the elements of $G$ (cosets of $H = 1$) is called the left regular representation of $G$.

 A: Let $G$ be a group, then you can get an homomorphisms 
$$ f \colon G \to S(G)$$
where $S(G)$ is the group of permutations on $G$, defined as the mapping that send every $g \in G$ in the mapping $f(g) \colon G \to G$ such that for every $x \in G$ the equation 
$$f(g)(x)=gx$$
holds.
Associativity and unit axioms grant that $f$ is an homomorphism of groups:


*

*for all $g,h \in G$ and $x \in G$ we have 
$$f(gh)(x)=(gh)x=g(hx)=f(g)(hx)=f(g)(f(h)(x))=f(g)\circ f(h)(x)$$

*for all $x \in G$ the equality
$$f(1)(x)=1x=x$$
holds.


This homomorphism $f$ is the left regular representation (it represents the elements of the group $G$ as symmetries of the same group seen just as a set).
The name left is to distinguish from the (anti)-homomorphism 
$$f' \colon G \to S(G)$$
that to every $g \in G$ associate $f'(g) \colon G \to G$ such that for every $x \in G$ we have $f'(g)(x) = xg$.
Via the correspondence between actions and homomorphisms in symmetric groups left regular representation are those representation that correspond to left-action of $G$ on itself given by multiplication.
Hope this helps.
