Math 25700 (33): Honors Algebra I
Homework 1 (Version 1)
Due: Monday, October 7, 2013
Note: Starred (*) problems are not to be turned in.
1. Read Chapters 0 and 1.
2. Let F be a eld. Consider the binary operation : F F F dened by a b = ab + a + b.
(a) Is
Math 25700 (33): Honors Algebra I
Homework 1 (Version 2)
Due: Wednesday, October 8, 2014
Note: Starred (*) problems are not to be turned in.
1. Read Chapters 0 and 1.
2. Let F be a eld. Consider the binary operation : F F F dened by a b = ab + a + b.
(a)
PROBLEM SESSION 9
Solutions for the second midterm exam:
1. False. A5 has order 60 but no element in A5 has order 15.
2. False. D16 has order 16 = 42 and is non-abelian.
3. True. One way to think about this is that any automorphism of V4 permutes
the thre
PROBLEM SESSION 10
HW 9 Problem 7: I found the following picture from
http:/www.metafysica.nl/groups/d2 patterns 3.html
Figure 1. Seven symmetry groups of friezes
There are 7 dierent pattern types. To determine all the possible pattern types,
we should rs
FINITE ABELIAN GROUPS
Main Theorem: Every nite abelian group A is isomorphic to a direct product of
cyclic groups of prime power order. These cyclic groups are uniquely determined
by A.
Example 1: For any natural number n, if n = p1 p2 pl is its prime fac
PROBLEM SESSION 7
Exercises 19 from Section 4.3 in D and F:
Let H act on K by conjugation: for any h H, we dene
h : K K
k hkh1
For any x K, the conjugacy class of x in H is equal to the orbit of x under this
action. Lets denote the orbit of x by O(x). Wha
PROBLEM SESSION 8
: In HW 7 Problem 11(e), I think what we need to show is that G = T
Notice that T G while O(2) is not a normal subgroup of G.
O(2).
Some remarks about direct products and semidirect products:
Fact 1: If H
G, K
G, G = HK and H K = cfw_e,
PROBLEM SESSION 6
HW 4 Problem 3: Given a group G, how to nd the commutator subgroup G ?
I dont know if there is any good way in general to solve this kind of problems.
The following observations may be useful:
G is the smallest normal subgroup of G such
PROBLEM SESSION 5
Something that we can try to consider when solving group theory problems:
When normal subgroups appear, try to consider their quotients.
When we need to show some quotient group G/H is isomorphic to some
other group, try to use the rst
PROBLEM SESSION 4
A remark about homework 2 Problem 15: There is a convenient way to
compute the expression of 1 when and are elements in some symmetric
group Sn .
For example, when is written as cycle decomposition:
= (a1 a2 an )(b1 b2 bm )
Then
1 = (
PROBLEM SESSION 2
Problem: Determine the order of the full symmetry group of a dodecahedron:
Solution 1. (This is presented by one of you during the problem session.) Suppose A
is one of the faces of this dodecahedron. There are |D10 | = 10 symmetries of
Math 25700 (33): Honors Algebra I
Homework 2 (Version 2, without the typos)
Due: Wednesday, October 15, 2014
Note: Starred (*) problems are not to be turned in.
1. Consider the symmetry group of the regular tetrahedron. Each symmetry can be thought of as
Math 25700 (33): Honors Algebra I
Homework 4 (Version 1)
Due: Wednesday, November 5, 2014
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 4.
2. Show that if G/Z(G) is cyclic, then G is abelian.
3. For each of the following groups G
Math 25700 (33): Honors Algebra I
Homework 5 (Version 1)
Due: Monday, November 4, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 4.
2. For each of the following groups G, nd the commutator subgroup G .
(a) S4
(b) A4
(c) D2n
3
Math 25700 (33): Honors Algebra I
Homework 7 (Version 1)
Due: Monday, November 17, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 4 and Section 5.1.
2. Prove that if x, y S4 and are distinct 3-cyles with x = y 1 , then the su
Math 25700 (33): Honors Algebra I
Homework 2 (Version 2)
Due: Monday, October 14, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Consider the group S8 , and let = (1348)(267) and = (12)(45783). Compute the following:
(a) and
(b) o() and
Math 25700 (33): Honors Algebra I
Homework 6 (Version 1)
Due: Monday, November 11, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 4.
2. (*) Show that for n 3, the alternating group An is generated by its 3-cycles.
3. Show tha
Math 25700 (33): Honors Algebra I
Homework 3 (Version 1)
Due: Monday, October 21, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapters 1, 2, and 3.
2. (*) Let : G H be an isomorphism of groups. Show that G is abelian if and only i
Math 25700 (33): Honors Algebra I
Homework 9 (Version 2)
Due: Wednesday, December 4, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 5.
2. (*) Prove that a nite abelian group is the direct product of its Sylow subgroups.
3. Le
PROBLEM SESSION 1
Some remarks about the denition of groups:
If (G, ) is a group, then we have the following:
1. The identity element is unique in G. That is, if e1 , e2 G such that for any
g G, both e1 g = g = g e1 and e2 g = g = g e2 hold, then e1 = e2
Math 25700 (33): Honors Algebra I
Homework 7 (Version 1)
Due: Wednesday, November 19, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 4 and Sections 5.1 and 5.4.
2. (*) Let G and H be groups.
(a) Show that G H H G.
=
(b) Show
Math 25700 (33): Honors Algebra I
Homework 9 (Version 1)
Due: Wednesday, December 4, 2013
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapter 5.
2. (*) Prove that a nite abelian group is the direct product of its Sylow subgroups.
3. Le
Math 25700 (33): Honors Algebra I
Homework 6
Due: Wednesday, November 12, 2014
1. (*) Read Chapter 4.
2. Show that for n 5, the only proper subgroup of Sn with index less than n is An .
3. Prove that if x, y S4 and are distinct 3-cyles with x = y 1 , then
Math 25700 (33): Honors Algebra I
Homework 3 (Version 1)
Due: Wednesday, October 22, 2014
Note: Starred (*) problems are not to be turned in.
1. (*) Read Chapters 1, 2, and 3.
2. For F a eld, we dene
a11
.
Mn (F) = A = .
.
an1
a1n
.
.
.
ann
aij F, 1 i,