I have some more questions that I need your help on. If you could just give me a hint, or perhaps tell me how to start them, that would be great.
ques. 21.17- Prove that if &: G to H is a homomorphism and a is an element of G, then the order of &(a)|the order of a.
ques. 22.9- Assume N is a normal subgroup of G. Prove that if the number of right cosets of N in G is a prime, then G/N is cyclic.
ques. 22.12- Prove that every element of Q/Z has finite order.
I hope I'm not bothering you to much with all of these questions. Thank you.
21.17 Recall: if |a|=n and if am=e then n|m (this small theorem was covered, for example, in the section about cyclic groups) . This tell us that the problem will be done if we show that (q(a))order of a=e. And the last claim is easy.
22.9. You are given that |G/N|=prime number. And the groups of prime order are cyclic - this is an easy exercise and we have covered it.
22.12.Every element in Q/Z is of type Z+(p/q), where p, q are integers, q>0 (I have used additive notation). Z+(p/q)+Z+(p/q)+...+Z+(p/q) (q times) is Z+p and this Z. So the order of this element is certainly less than q.