Assume that G is a group of order 63 that has two Sylow subgroups whose intersecion is non-trivial. Show that G has an element of order 21.
So by Sylow's theroem, I know that 63=3^2*7, that the sylow 7 subgroup is a normal subgroup of G with 6 order 7 elements. My sylow 3 subgroup is of order 9 and there can be one or 2 sylow three subgroups. Since I know that s7 (sylow 7 subgroup) is normal, I was thinking about trying to use this fact ot generate a cylic subgroup of order 21. Am I anywhere on the right path?