Math 720

Fall 2000

Homework 2

1. A trivial classification. Let G be a group, show that the following conditions are equivalent:

I. is either 1 or p, for some prime number p.

II. Z_p or is trivial

III. The only subgroups of G are G itself and the identity.

2. Let H and K be subgroups of G.

a) Show HK is a subgroup of G if and only if HK=KH.

b) Show that if H and K are of finite index in G and [G:K] and [G:H] are relatively prime, then G=HK.

3. Show that there is no nontrivial homomorphism from Q to Z.

4. Let G be a finite group and let p be the smallest prime dividing Show that if there is a subgroup H<G such that [G:H]=p then H is a normal subgroup of G.

5. Classify all groups of order p² where p is a prime number.

6. Let G be a group, we define the commutator subgroup of G to be the subgroup of G generated by elements of the form x^-1y^-1xy with

a) Show that G¹, the commutator subgroup of G is a normal subgroup of G.

b) Show that G/G¹ is an abelian group.

c) Let be a homomorphism of groups with A abelian. Show that G¹<

d) Show for every automorphism f of G, f(G¹)<G¹. (Such a subgroup is called a characteristic subgroup).