Hey im preparing for my second exam and I need major help with these problems. I would greatly appreciate any help, hints or answers, that could help me understand how to do these problems. So that I can do well thanks!!!

G denotes a group

1. Let |G|=p*g*r with different primes p, g, r, and let H,K<=G with |H|=p*g, |K|=g*r. Show:|H (intersection) K|=g.

2. Let H be the normal subgroup G, a in G and |H|=10. The element aH of the group G/H has order 3; what are the possibilities for |a|? Explain.

3. Let G be a group and G' be the subgroup generated by S={x^-1*y^-1*x*y| x,y in G}. Then:

a. Prove that G' is normal in G.

b. Prove that G/G' is abelian.

c. If G/N is abelian, prove that G'<=N.

d. Prove that if H<=G and G'<=H then H is a subgroup of G.

4. Suppose that G is an abelian group of order 120 and that G has exactly 3 elements of order 2. Determine the isomorphism class of G. Explain.