Let G have order pq, where p and q are distinct primes. Prove G is either cyclic or every element x≠e in G has order p or q
One corollary of LaGrange's Theorem is that the order of every element in G is a divisor of the order of G, which means it is either order p or order q, since those are the only divisors of |G|. The only other possibility for order of elements is that an element can have order pq, which by definition means it is cyclic. (not so sure on this part, I think that is correct though).
Yes. I do beleive that is correct.
Problem: Let have order , where are distinct primes. Prove is either cyclic or every element has order or
Proof: As was stated we know that if that . Now suppose that , we must have that and since are distinct primes this is only true if . If equals the first two we are done. Otherwise we'd have that which means that are distinct elements. If there existed a such that we'd have that which is clearly false. Thus .