Finding permutations $\sigma\in S_4$ satisfying $\sigma^2=(1\;2)(3\;4)$

Question

I am struggling to find the answer to the following question: Find all $\sigma\in S_4$ satisfying
$σ^2 = (1\;2)(3\;4)$

Can somebody please explain the way to find these? Thank you!

Answer 1

Hint: To be consistent, we always write the smallest number in the cycle first, in this case $1$.

What type of shape could $\sigma$ take? Could it be a 3cycle? Could it be two 2cycles? Must it be a 4cycle?

Supposing it were a 4cycle, we begin writing it like so: $(1~*~*~*)$. Since $\sigma^2=(1~2)(3~4)$, what does that imply about where $2$ should be located in the above?

Answer 2

In practice, calculating the $k$-th power of a cycle consists in jumping from one element to the $k$-th next element (position calculated modulo the cycle length).

So for a cycle $\gamma$ such that $\gamma^2=(1\,2)(3\,4)$, we know that $\gamma$ must have length $4$, and further that $\gamma=(1\,x\,2\,y)$, $x,y\in\{3,4\}$. So there are two solutions: $$\gamma=(1\,3\,2\,4)\quad\text{and}\quad \gamma^{-1}=(1\,4\,2\,3). $$