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 $\det(B) = -1$.

  • $\begingroup$ Maybe you should clarify your definition of delta function.(Also $K$ should be of other dimension?) Also you are talking of the function which doesn't sound like a set to me and the permutation matrices. What you should do is look at the definition of permutation and what the permutation matrices do on the standard basis. I think it should be this way: A permutation matrix is given by such a Kronecker Delta and you want to show that those matrices are the permuations? $\endgroup$ – user60589 Nov 3 '16 at 14:52
  • $\begingroup$ @user60589 Yeah K should be of another dimension, it was just an example. The delta function definition was just the definition of kronecker according to wikipedia. And yeah you're right about the question, am I on the right track? $\endgroup$ – Andrew Raleigh Nov 3 '16 at 15:26
  • $\begingroup$ Maybe you can explain how the Kronecker Delta function is related to $S_7$ or $P$? And you should write down $\phi$ and then prove the things about $\phi$ right? $\endgroup$ – user60589 Nov 3 '16 at 15:44
  • $\begingroup$ It's defined as $\phi:S_7 \to P$ 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 matrix with $ij$-entry being $\delta_{i, \alpha(j)}$. $\endgroup$ – Andrew Raleigh Nov 3 '16 at 16:25
  • $\begingroup$ Right you wrote it also in your question. What I wanted to say is that the delta function is a nice way to write your matrix. For one-to-one you need to take two permutations and show that both resulting matrices are different. $\endgroup$ – user60589 Nov 8 '16 at 11:50

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.


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