Math 417: Midterm 1 Study Guide
1
Outline
1. Groups
(a) Definition 1.10
(b) Examples
i. Symmetries of Polygons and Polyhedra 1.2, 1.3, 1.4
ii. Permutations 1.5
iii. Zn under addition 1.6, 1.7
iv. Othe
Homework 8
2.7.4. Claim: Suppose G is a finite group. Let N be a normal subgroup of G and A an
arbitrary subgroup. Then
|A|N |
|AN | =
.
|A N |
Proof: Suppose G is a finite group. Then by Lagranges Th
Name:
Math 417: Midterm Exam 1
Section A: Short Answer
In this section, you must provide the answer to the question only, you are not required
to give a proof. Each question is worth 10 points.
A1. [1
Homework 4
1.9.1. Claim: For any x, y, and n N,
n
(x + y) =
n
X
n
k=0
k
xk y nk .
Proof: We will prove
the above equation by induction on n. Suppose first that n = 1.
Then (x + y)1 = 10 x + 11 y, so
Homework 6
2.3.6. Claim: The subgroup H defined below of D6 is isomorphic to D3 .
Proof: The group D6 is the symmetry group of the hexagon. Let a be a flip over a line of
symmetry of the hexagon, and
Name:
Group Vocabulary Quiz
Fill in the blanks with the appropriate term from the box.
left coset
normal
image
S3
right coset
homomorphism
D4
A3
1. A(n)
distributive
isomorphism
GL(2, R)
S6
abelian
en
Homework 1
1.3.1. Claim: There are 6 symmetries of an equilateral triangle. Three are achieved by
rotating about the line from a vertex to the center of the opposite edge. Three can be
described as ro
Name:
Group Vocabulary Quiz
Fill in the blanks with the appropriate term from the box.
left coset
normal
image
S3
right coset
homomorphism
D4
A3
distributive
isomorphism
GL(2, R)
S6
abelian
endomorphi
Homework 3
1.6.9. Claim: If p is prime and p divides a product a1 ar of nonzero integers, then p
divides one of the factors ai .
Proof: We will proceed by induction on r. As a base case, suppose r = 1
Math 417: Midterm 2 Study Guide
1
Outline
1. Chapter 2
(a) Homomorphisms
i. Kernel
ii. Image and Inverse Image
iii. Isomorphism Theorems
(b) Subgroups
i.
ii.
iii.
iv.
Subgroup lattice
Cosets and Lagra
Homework 7
2.5.7 Claim: If N G then the following are equivalent.
(a) N is normal.
(b) Each left coset of N is also a right coset.
(c) For each a G, aN = N a.
Proof: First we will show that if N is no
RSA Encryption.
RSA encryption is a type of what is called public key encryption. Public key encryption works as follows. We start with some mathematical operation which is easy to do,
but impossible
Homework 5
2.1.5. Claim: If G is a group such that all elements g G satisfy g 2 = e, and H is
isomorphic to G, then H also has this property. Therefore Z4 is not isomorphic to the
group of symmetries
Name:
Math 417: Midterm Exam 1
Section A: Short Answer
In this section, you must provide the answer to the question only, you are not required
to give a proof. Each question is worth 10 points.
A1. [1
Name:
Math 417: Midterm Exam 2 Practice
Answer the following questions in the space provided. You may use the backs of pages for
scratch work, but it will not count towards your mark. If a questions b