Generators of $S_4$

Question

The group $S_4$ is generated by $\{(12), (1234)\}$.Now something which I want to know is that how will I generate by a cycle of order $4$ and any cycle of order $3$ and order $2$.

My main question is to find the number of homomorphisms, and if $\varphi(1234) \to -1$ and $\varphi (12) \to -1$.

Then how will I conclude from here that any odd permutation will go to $-1$ and any even will go to $1$. (It has to be shown by crude calculations and not by using homomorphism theorems).

I really need some help.

Answer 1

Any odd permutation $π$ must be expressible as a word in $(12)$ and $(1234)$ of odd length $n$, since both are odd. Thus $\varphi(π)=(-1)^n=-1$, since $\varphi$ is a homomorphism.

Answer 2

Using a representation theory (see this) I've been working, I developed a string processing python program to multiply permutations. Step by step I increased the size of our working group subset; there is one and only one standard representation.

$\tau = (12)$

$\sigma = (1234) = (12)\,(23) \,(34)$

$\sigma^2 = (13)\,(24)$

$\sigma^3 = (14)\,(24)\,(34)$

$\tau\sigma = (23) \,(34)$

$\tau\sigma^2 = (13) \, (24) \, (34)$

$\tau\sigma^3 = (14)\,(34)$

$\sigma\tau = (13) \,(34)$

$\sigma^2\tau = (14) \, (23) \, (34)$

$\sigma^3\tau = (24)\,(34)$

$\tau\sigma\tau = (13) \, (23) \,(34)$

$\tau\sigma^2\tau = (14) \, (23)$

$\tau\sigma^3\tau = (12) \, (24) \,(34)$

$\sigma^2\tau\sigma^2 = (34)$

From here you can 'peel off' the remaining $5$ transpositions and explain why, for any transposition $\omega$,

$\quad \varphi(\omega) = -1$