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?

This answer is not a theoretical approach but it is base on using GAP. By using the following codes in GAP environment; you can easily find the desired elements in second equation:

S_7:=SymmetricGroup(7);
e:=Elements(S_7);;
for i in [1..Size(e)] do if e[i]^3=(1,2,3,4) then Print(e[i],"\n"); fi; od;



(1,4,3,2)
(1,4,3,2)(5,6,7)
(1,4,3,2)(5,7,6)

