Prove that a group of order 42 has a nontrivial normal subgroup

[Viking85]

Homework Statement

Prove that a group of order 42 has a nontrivial normal subgroup

Homework Equations

We are supposed to use Cauchys Theorem to solve the problem
We are not allowed to use any of Sylows Theorems

The Attempt at a Solution

By using Cauchys Theorem i know there there is an element of order 7
I then let A = <a> where o(a)=7
Essentially i know the normal subgroup is of order 7, but i need to still show it is normal

 

[Dick]

If A isn't normal then it has a conjugate subgroup B which also has seven elements. What's the order of AB?

 

[Viking85]

Well, I think that A ∩ B = 1 or 7 (Using Lagranges Theorem)
If A ∩ B is 1 it results in a contradiction because that would imply ord(AB)=49 > ord(G)

Which means that A ∩ B = 7

Which means the A = B ??

 

[Dick]

Well, I think that A ∩ B = 1 or 7 (Using Lagranges Theorem)
If A ∩ B is 1 it results in a contradiction because that would imply ord(AB)=49 > ord(G)

Which means that A ∩ B = 7

Which means the A = B ??

Right. That makes A normal, yes?