Homework 7
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due this Friday, March 22.
1. Let G =
1a
01
aZ .
(a) Prove that G is a subgroup of GL(2, R).
(b) Prove that G is isomorphic to Z.
2. Let G, H , and K be groups, and let
Takehome Final
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due next Tuesday, May 21.
Rules: This is a nal exam, not a homework assignment. You must solve the problems
entirely on your own, and you should not discuss the pro
Homework 5
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due this Friday, March 8.
1. (a) List the possible cycle structures for an element of S6 .
(b) Make a table showing the number of elements of S6 with each cycle structu
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
March 1, 2013
Homework 4 Solutions
Problem 1.
(a) The following table shows the isomorphism type of each of these groups:
Group U (3) U (4) U (5) U (6) U (7) U (8)
Type
Z2
Z2
Z4
Z2
Z6
V
(b) The orders of the
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
February 22, 2013
Homework 3 Solutions
Problem 1.
Let G =
ab
01
a, b R and a = 0 .
(a)
Proposition. G is a subgroup of GL(2, R).
Proof. Note rst that G is nonempty. Note also that G GL(2, R), since every elem
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
February 14, 2013
Homework 1 Solutions
Problem 1.
(a) Consider the following matrix, whose entries are elements of the eld Z11 .
A=
74
.
43
The characteristic polynomial of this matrix is:
7
4
= (7 )(3 ) 16 =
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
February 15, 2013
Homework 2 Solutions
Problem 1.
Let G = (0, ) R, and let be the binary operation on G dened by
(a, b) (c, d) = (ac, bc + d)
for all (a, b), (c, d) G.
Proposition. G forms a group under the o
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
March 8, 2013
Homework 5 Solutions
Problem 1.
The following tables show the number of elements of S6 with each cycle structure:
Cycle Structure
Number
e
1
( )
15
( )
40
( )
90
( )
144
( )
120
Cycle Structure
Homework 6
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due this Friday, March 15.
1. Let G be a group. Let H be a subgroup of G, let x G, and let
xHx1 =
xhx1 | h H .
(a) Prove that xHx1 is a subgroup of G.
(b) Prove that xH
Homework 10
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due this Friday, May 3.
1. Let G be a group, let H and K be subgroups of G, and let a G. Prove that
(aH ) (aK ) = a(H K ).
2. Let G be a group, let H and K be nite sub
Homework 11
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due this Friday, May 10.
1. Let G = U (5) U (5), and let N = cfw_(1, 1), (2, 3), (3, 2), (4, 4).
(a) List the elements of the four cosets of N in G.
(b) Determine the
Homework 8
Math 332, Spring 2013
A
These problems must be written up in L TEX, and are due this Friday, April 5.
1. (a) Find an eight-element subgroup G of GL(2, Z3 ) containing the matrices
and
02
10
11
.
12
(b) What is the isomorphism type of G?
2. Let
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
April 11, 2013
Homework 7 Solutions
Problem 1.
Let G =
1a
01
aZ .
(a)
Proposition. G is a subgroup of GL(2, R).
Proof. Clearly G is a nonempty subset of GL(2, R. Next, if
1a
01
1
=
1a
G, then
01
1 a
.
0
1
Si
Math 332: Abstract Algebra
Prof. Belk
Jim Belk
April 11, 2013
Homework 6 Solutions
Problem 1.
Let G be a group. Let H be a subgroup of G, let x G, and let
xHx1 =
xhx1 | h H .
(a)
Proposition. The set xHx1 is a subgroup of G.
Proof. Since H is a subgroup,
Dynamical Systems, Fall 2012- Final Project Linh Pham, Shuyi
Weng
1. Introduction: Fibonacci Word Fractal In class we talked about
self -similar fractal curve, so in this project we will explore a curve which
is also self-similar fractal. That is called F