If $X$ is a matrix representation fo a group $G$, then its kernel is the set $N=\{g\in G: X(g)=I\}$. Now I read Sagan's book, GTM 203, the Symmetric group. Now I have some problems in the exercises of Chapter 1.
If $X$ is a matrix representation fo a finite group $G$, then its kernel is the set $N=\{g\in G: X(g)=I\}$. Define a function $Y$ on the group $G/N$ by $Y(gN)=X(g)$ for $gN\in G/N$.
I have proved that
(a). N is a normal subgroup of G and for the coset representation, $N=\cap_{i}g_{i}Hg_{i}^{-1}$ where the $g_{i}$ are the transversal.
(b). $Y$ is a well-defined faithful representation of $G/N$, $Y$ is irreducible if and only if X is.
Now I have no idea for this question:
(c). If $X$ is the coset representation for a normal subgroup $H$ of $G$, what is the corresponding representation $Y$?
This problem has bothered me for many days. I would appreciate any help! Thank you very much!
 A: If $H$ is normal then by part (a) the kernel of the coset representation $X$ is just $H$ itself, so $Y$ is indeed a representation of $G/H$.
I am not sure how your book formalizes the coset representation, but one way is taking a free vector space $V$ on the set of cosets $G/H$; that is, we have a basis whose elements are the cosets of $H$ in $G$, and elements in the vector space are just formal sums of these basis elements. For instance, $3(g_1H) + 7 (g_2H)$ could be one vector in this space. Addition is defined componentwise, so $$(3(g_1H) + 7(g_2H)) + (4(g_1H)) = 7(g_1H) + 7(g_2H).$$ Thus the zero vector is $0(g_1H) + \dots + 0(g_nH)$ where the $g_i$ are a transversal.
Then $X$ can be thought of as a homomorphism $G\to GL(V)$, and we can define $X(g)$ by defining how it acts on a basis of $V$. For the coset representation, we want $X(g)(aH) = (ga)H$ for any $g,a\in G$. If you would like to think of this as a matrix representation, then you can just identify $V$ with $\mathbb{C}^{|G:H|}$, with an isomorphism given by mapping $g_iH$ to $e_i$ (the $i$th standard basis vector) for each $g_i$ in a transversal set.
The point is that now $Y$ is a homomorphism $G/H\to GL(V)$ such that $Y(gH) = X(g)$. So In particular $Y(gH)(aH) = (ga)H$ for any $g,a\in G$. This is exactly the left regular representation of $G/H$.
