A permutation matrix is a square matrix with only a single $1$ in each row and each column, with the rest being $0$s. Here's an example:
$$K = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 &0 & 0\\0&0&0&1\\0&1&0&0 \end{bmatrix} $$
Meanwhile a Kronecker Delta function is defined as $$\delta_{i, \alpha(j)} = \begin{cases} 1 & \text{if } a = b \\ 0 & \text{if } a \ne b\end{cases}$$
What I'm trying to prove is that there exists an isomorphism from $\phi:S_7 \to P$ where $P$ is the set of $7 \times 7$ permutation matrices. And if $\alpha$ is a permutation in $S_7$, then $\phi(\alpha)$ is the matric with $ij$-entry being $\delta_{i, \alpha(j)}$.
I understand that in order to prove an isomporphism, I have to prove that $\phi$ is one-to-one, onto, and operation preserving.
| one-to-one
The Kronecker Delta function would resemble an Identity matrix, and the identity matrix can also be in the set of permutation matrices. So, it is one-to-one.
| onto & OP
I'm afraid I don't get either of these. As I understand it, onto implies if you can go from one way you should be able to go back from that way. It seems true but I don't understand how I would go about it.
As for OP, there should be variables $x, y \in G$, and we need to show that $\phi(xy) = \phi(x) \phi(y)$. The issue I have here is that as far as I see, there is no operation for me to test it out with.
Any help with proving that it's an isomorphism would be great. Also, why is it that if $\beta \in S_7$ then $\det(\phi(\beta)) = \operatorname{sgn}(\beta)$? I know that since $\beta$ is the same as the identity matrix except that two rows have been switched, then $\det(B) = -1$.
The permutation group $S_n$ is defined as the bijections of a $n$-element set. Most times you fix an set like $\{1, \dots,n\}$ that it is properly defined.
Now you should look what your permutation matrices do to the standard basis $e_1 \dots, e_n$, where the standard basis is defined by $e_1= (1,0, \dots, 0)^t, \dots$.
You should find out that the set $\{e_1, \dots, e_n\}$ is closed under the action by the permutation matrices. In fact every permutation matrix gives a bijection on this set.
Now it is easy to see that there is a nice bijection $\xi$ of $\{1, \dots,n\}$ and $\{e_1, \dots, e_n\}$.
Then the inverse of $\phi$ is given by $\psi \colon P_n \to S_n, K \mapsto (n \mapsto \xi^{-1}( K \xi(n)) )$. You can either check that this map is inverse to $\phi$ or show that $\phi$ is injective and surjection. What you also could to is only show injectivity and show that both sets have same cardinality.