First: show that .
Secondly: Observe that
Let and show that
Let and be subgroups of the group , and let . Show that either or else is a left coset of .
I've been working on this one for over a day, now, and I'm still stumped. Assistance would therefore be much appreciated!
EDIT: I finally solved it, and it is indeed extremely simple. I can't believe I missed it for so long.
Proof: Suppose , and let . Then there are , with . So , and therefore . The conclusion follows.
To show that is quite straightforward: Let . Then there are with . So , and therefore . Now let . Then there is with and therefore . So . The conclusion follows.
However, I am unable to show that .
If anyone has a solution, or the outline of a solution, it would be much appreciated!
Perhaps my last post didn't really clarify that much:
I hope this will help:
Observe that . This is easy to prove.
One can show that .
This follows from the fact that different cosets are disjunct. (for any , we have )
I'll try to give a better explanation later.
It helps to recall that cosets partition a group, which I had forgotten. Since or implies is or , respectively, then we can assume , which means
as you have already noted. However, I don't see how this implies . I worked on it for several hours this morning without further progress.