Hello! I need help with showing the proof for part b. I have done the beginning part of the proof of assuming p=q. However, I don't know how to show the other case. Is it possible to prove this part of the proof without assuming that G is simple, meaning that G is a nontrivial group whose only normal subgroups are the trivial group and the group itself? If possible, please show the steps on how to prove the other case for part b without assuming G is simple.

Problem 3. Let p and q be primes. a.) Suppose that G is a group of order pq. Show that G has a nontrivial normal subgroup, i.e. a normal subgroup N such that N 79 {e} and N 79 G. (Hint: Don't forget to consider the case p = q. You might nd the result of problem 4b on Hw S useful in this case} b) Suppose that G is a group of order $3). Show that G has a nontrivial normal subgroup. (Hint: Be careful to consider all cases: p q and p = q. In this last case you ma},r nd the result of problem 5b on Hw 8 useful.}