Math 546
Problem Set 24
1. Suppose that H is a normal subgroup of the group G and G / H is Abelian.
Show that for every a,!b !G , aba !1b !1 "H .
Hint: Consider the product of the corresponding left cosets.
2. Suppose that ! : Z 20 " Z 33 is a homomorphism. What is ! ?
Solution: ! must be the identity mapping.
3. How many automorphisms are there for the group Z 30 ?
Solution: There are 8 automorphisms.
4. Suppose that ! :U(40) " U(40) is a homomorphism and ker ! = {1,!9,!17,!33} .
Describe the set of all elements of U(40) that map to 11.
Solution: It is possible that nothing maps to 11. However, suppose that ! (a) = 11 .
Then any element of aK = {a,!9a,!17a,!33a} also maps to 11 (where the
multiplication is in U(40) .
5. (a). Suppose that Z12 is a homomorphic image of the group G.
What can you say about the order of G?
Solution: The order of G is a multiple of 12.
(b). Suppose that for every prime number p, Z p is a homomorphic image
of the group G. What can you say about the order of G?
Solution: G must be infinite.
(c). Give an example of a group described in (b).
6. Suppose that ! : S3 " G is a homomorphism. Explain why it is that if we
know the value of ! on each of (1,!2),!(1,!3), and (2,!3) , then we know all
the values of ! .
7. Prove: Every Abelian group of order 10 is cyclic.
Hint: Use Cauchy’s Theorem and find a candidate for a generator of the group.
Then verify that the element you picked really does generate the group.