In $S_7$, $x^2=(1234)$ has no solutions but $x^3=(1234)$ has at least two How would I go about showing that in $S_7$, the equation $x^2=(1234)$ has no solutions but the equation $x^3=(1234)$ has at least two solutions?

You can show that the second equation has at least two solutions by finding examples.  
You can show that there is no $x$ satisfying the first equation by considering what the order of $x$ would be, and what the possible orders of elements of $S_7$ are.
