Find the smallest subgroups of all permutations of $8$ elements which contain specific permutations [duplicate]
That's exactly right. In any case, to find the smallest subgroup containing a single element, you simply take all the powers of that element. It will be a finite group if and only if some power of the element is equal to the identity.
To prove that the powers of $x$ are the smallest subgroup containing $x$, first it is clear that any subgroup that contains $x$ must also contain all powers of $x$. Conversely, all powers (both positive and negative) or $x$ form a subgroup. So we have shown that both sides are included in the other, and so they must be equal.