Hi, resident911.

Some tips:

Problem #1:

Define

Since G is a p-group, a consequence of the orbit-stabilizer theorem is that

mod p.

Hopefully this looks familiar from class or a book. If not I can give more details later.

Now use our assumption on the order of X and see what you can do.

Problem #2:

I would suggest looking at the Sylow theorems and seeing where they take you.

Give it some thought and I'm sure you'll get it!

Let me know if there's still confusion. Good luck!