Abstract Algebra - Normal groups

[TheForumLord]

Homework Statement

I'll be delighted to receive some guidance in the following questions:

1. Let G1,G2 be simple groups. Prove that every normal non-trivial subgroup of G= G1 x G2 is isomorphic to G1 or to G2...

2. Prove that every group of order p^2 * q where p,q are primes is solvable...

Homework Equations

The Attempt at a Solution

I've no idea about the second question. But the first one seems very easy, but I can't figure out how to solve it.


Help is needed

Thanks in advance

 

[mighty2000]

I'm happy to assist you with these questions. For the first question, we can use the fact that if H is a normal subgroup of G1 x G2, then H is a direct product of normal subgroups of G1 and G2 respectively. Since G1 and G2 are simple groups, their only normal subgroups are the trivial subgroup and the group itself. Therefore, the only possible non-trivial normal subgroups of G1 x G2 are direct products of G1 and G2. Can you take it from here and prove that these are indeed isomorphic to G1 or G2?

For the second question, we can use the fact that every group of order p^2 * q must have a normal subgroup of order p^2. This is because by Sylow's theorems, there must be a subgroup of order p^2, and since p is prime, this subgroup must be normal. Can you think of how to use this normal subgroup to show that the group is solvable?

I hope this helps. Let me know if you need any further clarification. Good luck with your homework!