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!