Department

MathematicsCourse Code

MATH 546Professor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**2 pages of the document.**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→G′be 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”.)

###### You're Reading a Preview

Unlock to view full version