Math 441 Algebra                                     Hw 18                                                 Name:

Due: October 28

1. A Group is simple if it has no proper nontrivial normal subgroups.

Show that if a finite group G contains a nontrivial subgroup of index 2 in G, then G is not simple.

2. Classify the given factor group.

   (a)  

   (b)

   (c)  

3. Let  be a group homomorphism.

(a)    If  is a normal subgroup of  then  is a normal subgroup of .

(b)   If  is a normal subgroup of , then  is a normal subgroup of .

4. Prove: Let H be a normal subgroup of a group G.

(a)    If G is abelian, then G/H is abelian.

(b)   If G is cyclic, then G/H is cyclic.