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. Other examples: Z, R, C under addition, GL(n, R)
(c) Proper
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 Theorem, |G/H| = |G|/|H| for
any normal subgroup H of G.
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. [10 points] Let G be a nonempty set, with a multiplicatio
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 the equation holds. Now suppose the equation holds for
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 r be a rotation counterclockwise by /3 radians. Let e b
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
endomorphism
SL(2, R)
A6
group is commutative.
2. The gro
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 rotations of the vertices, by 0, 2/3, 4/3 radians. The mu
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
endomorphism
SL(2, R)
A6
associative
kernel
Z6
D8
1. A(n) abelian
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. Then p divides
a1 , hence p divides a factor. Suppose
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 Lagranges Theorem
Normal subgroups
Quotient groups
2. Chapte
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 normal, then each left coset of N is also a right coset
o
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 to efficiently undo without some extra information. The
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 of a rectangle.
Proof: Suppose G is a group such that f
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. [10 points] Let G be a nonempty set, with a multiplicatio
Homework 2
1.5.3.
1 2 3 4 5 6 7
(a)
= (1257)(364).
2 5 6 3 7 4 1
(b) (12)(12345) = (2345).
(c) (g) (12)(13)(14) = (1432).
(d) (h) (13)(1234)(13) = (1432)
1.5.6. Claim: We can compute the inverse of a permutation in two line notation by switching the top a
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 begins with the word
prove or show then a proof is requi