Show that a group of order 70 can not be simple.
I've tried to solve using Sylow theorem. I got 1, 5, 7, 35 Sylow 2-subgroups, 1 sylow 5-subgroup and 1 sylow 7-subgroup. Now the only choice is 35 Sylow 2-subgroups which would yield 36 elements. Now we are left with 34 elements but we have only one sylow 5-subgroup and one sylow 7-subgroup.
Why all the elements of sylow subgroups are not adding up to 70?