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$Let \ \alpha=(1235).$ Indicate all values of $n \in \mathbb{N}$ in which $\alpha^n \in A_7$ and indicate all values of $m \in \mathbb{N}$ for which there are a permutation in $S_7$ with order $m$.

I'm having trouble getting the values of $n$. For the second question I got $m$={1,2,3,4,5,6,7} but even this I'm not sure.

Note: $\alpha$ is a permutation in $S_7$

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    Do you know the definition of $A_n$? – 2017-01-17
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    It represents the even permutations – 2017-01-17
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    And is $(1235)$ even or odd? – 2017-01-17
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    Dont know why are you being so rude but okay. Its odd. – 2017-01-17
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    I'm not trying to be rude, my apologies if I came across as so. I'm trying to lead you to the answer as opposed to just telling you; I do not think that would be as beneficial – 2017-01-17

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The order of an element of $S_n$ is the least common multiple of the size of its cycles. The lengths of the cycles of an element of order $m$ must add up to $n$. If the lengths of the cycles are $\ell_1,\dots,\ell_k$ then $\mathrm{lcm}(\ell_1,\dots,\ell_k)=m$ and $\ell_1+\dots+\ell_k=n$.

In $S_7$, try the example $(123)(4567)$. Does it really have an order $m\in\{ 1,2,\ldots ,7\}$?