Math 546, Final Exam , Fall 2004
The exam is worth 100 points.
Write your answers as legibly as you can on the blank sheets of paper provided.
Use only one side of each sheet. Take enough space for each problem. Turn in
your solutions in the order: problem 1, problem 2, ... ; although, by using enough
paper, you can do the problems in any order that suits you.
I will grade the exams on Saturday. When I finish, I will e-mail your grade to
you.
I will post the solutions on my website when the exam is finished.
1. (7 points) STATE and PROVE Cayley’s Theorem.
2. (7 points) Apply the proof of Cayley’s Theorem to the element (1,2,3) of the
group
A4={(1),(1,2,3),(1,3,2),(1,2,4),(1,4,2),(1,3,4),(1,4,3),(2,3,4),(2,4,3),
(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)}.
(Number the elements of A4using the order I in which I listed the elements.)
What do you get?
3. (7 points) Let ϕ:G→G0be a group homomorphism. Prove that ϕis
one-to-one if and only if the kernel of ϕis {id}.
4. (7 points) Give an example of a non-abelian group of order 16. A very short
explanation will suffice.
5. (7 points) Give an example of an abelian, but non-cyclic, group of order 16.
Explain.
6. (7 points) Let Hbe the subgroup <(1,2,3)>of the group G=A4,and
let Sbe the set of left cosets of Hin G. Define multiplication on Sby
(g1H)(g2H)=(g
1
g
2
)Hfor all g1and g2in G.IsSa group? Explain very
thoroughly.
7. (9 points) Let Nbe a normal subgroup of the group Gand let Hbe any
subgroup of G. Let HN be the subset {hn |h∈Hand n∈N}of G.
(a) Prove that HN is a subgroup of G.
(b) Prove that Nis a normal subgroup of HN .
(c) Let ϕ:H→HN
Nbe the group homomorphism which is given as the
composition of inclusion H→HN , followed by the natural quotient map
HN →HN
N. What is the kernel of ϕ?
(d) Apply the First Isomorphism Theorem to ϕ.
(You just proved the “Second Isomorphism Theorem”.)
