Hi guys.

I need to prove the following statment, I did most of the work (which I'm not entirely sure is valid - so a feedback would be great), but I'm having some trouble proving the last part of it, and could really use some tips here.

Let be a group.

Let be non-trivial subgroups of ( ).

Prove that .

Proof:

If we're done, since , and , we get that there exist s.t .

If , without loss of generality, suppose that .

Let s.t (there exist such since isn't trivial and ).

Let .

Obviusly , so If we show that , we're done.

Let's assume by contradiction that .

It then follows that .

Case ( ):

If , then , for some .

But then:

but ! that contradics our assumption of .

Case ( ):

I have nothing here...

any help would be greatly appreciated!