SYLLABUS
Education 55
Knowing and Learning in Mathematics and Science
Fall, 2016
School of Education
University of California, Irvine
Instructor
Office Hours
Rossella Santagata, Ph.D.
[email protected]
By appointment
Education 3457
COURSE INFORMATION
Cl
120A Hw4
Zian Deng
20211966
Section 5
22.
The elements of this cyclic group are
30.
[cos(4/5) + i sin(4/5)]^k = 1
cos(4k/5) + i sin(4k/5) = 1
k=5n/2
As k is an integer, n=2
Thus k=5, the order is 5
36,
a.
+
0
1
2
3
4
5
0
0
1
2
3
4
5
1
1
2
3
4
5
0
2
3
4
5
120A Hw5
Zian Deng
20211966
Section 8
4
1 2 3 4 5 6
5 1 6 2 4 3
7
(1,4)(2,3)
Thus the answer is 2
12. cfw_1,2,3,4
16. 3!=6
44.
An n-gon has n rotational symmetries and n reflection symmetries, thus the order is 2n
Suppose the first vertex is (r, ), second
120 HW9
Zian Deng
20211966
Sec14
6. As a subgroup of Z12 X Z18, <(4,3)> has an order of 6, the order of the given factor group is
12*18/6=36
7. As <(1,p1)> has an order of 6, the order of the given factor group is 2*6/6=2
12. < (1, 1) >= cfw_(1, 1),(2, 2)
120 HW10
Zian Deng
20211966
Sec15
13
The center of D4 can be computed from Table 8.12 by looking for elements with the same row sequence
as column sequence.
So Z(D4) = cfw_0, 2.
Recall that the center of a group is always a normal subgroup.
Note that D(4)
120A Hw6
Zian Deng
20211966
Section 9
29If all the elements of H are even, then we are done.
If not, H contains at least one odd permutation, we call it h.
Let T be the map that right multiplies by h, for which is T(x) = xh. Since H is a subgroup,
multipl
120A HW7
Zian Deng
20211966
Sec 10
19
a.T
b.T
c.T
d.F
e.T
f.F
g.T
h.T
i.F
j.T
24Every coset has the same number of elements, but 4 is not a divisor of 6, that is to say we
cant divide 6 elements into 4 cells with each cell having same number of elements.
Math 120A Dis 10
Quiz 6
Fall 2011
Solution
1) (3 pts.) Let : G H be a homomorphism of groups. Define the kernel of .
The kernel of is the subgroup of G defined by ker g G ( g ) eH .
3 points for a specification of the set.
2) (3 pts.) Find all subgroups o
Math 120A Dis 10
Quiz 8
Fall 2011
Solution
1) (2 pts.) Let G be a group and H be a normal subgroup of G . Define the factor
group of G by H .
The factor group of G by H is the set of left cosets of H in G (denoted G / H ) under
the operation aH bH ab H .
120A Hw2
Zian Deng
20211966
Section 3
8 No, it is not isomorphism because it is not 1 to 1. Two different matrices may
have same determinant.
11. No, it is not isomorphism because it is not 1 to 1. Two different functions may
have same derivative.
12. No,
120A HW 1
Zian Deng
20211966
1.2
1.
bd=e
cc=b
[(ac) e] a=a
3. (bd) c=a
b (dc)=c
Not associative as (bd)cb(dc)
5.
*
a
b
c
d
a b c d
a b c d
b d a c
c a d b
d c b a
10. ab=2^(ab)=2^(ba)=ba
(ab) c=2^[2^(ab)c]
a (bc)=2^[2^(bc)a]
Thus it is commutative but not
Math 120A Homework 5
Hand in questions 1, 2, 4, 5, 7 and 9 at the discussion class on Tuesday 17th November
1. Find all the cosets of the following subgroups: since the groups are Abelian, left and right cosets
are identical.
(a) 4Z 2Z
(b) h4i Z10
(c) h6i
Math 120A Homework 1
Denitions Complete the following sentences.
1. A binary relation on a set X is associative if and only if
2. A binary relation on a set X is commutative if and only if
3. The table of a commutative binary relation has
(type of symmetr
Math 120A Homework 2
Hand in questions 1(a), 2(e), 3(b), 4, 5(b), 8, 9(a,b,c) & 11 at the discussion on Tuesday 13th October
1. Prove directly that the following are structural properties: i.e. if is an isomorphism of binary
structures, then each property
Math 120A Homework 6
Submit questions 1, 2, 3, 5, 7 and 11 at the discussion class on Tuesday Dec 1st.
Question 12 onwards are extra questions to help you prepare for the nal.
Factor Groups
1. Let : Z18 Z12 be the homomorphism with (1) = 10.
(a) Find the
Math 120A Homework 7
Nothing for submission: just extra questions on group actions for the nal
1. In Sn let Hj = Stab( j) (1 j n) be the isotropy subgroup of j cfw_1, 2 . . . , n. By exhibiting
an element Sn such that ( j) = k, show that Hj Hk for all j,
Math 120A Introduction to Group Theory: Final Exam Spring 2009
Total marks = 100
1. Find all the homomorphisms : Z10 Z15 . Make sure you explain how you know youve found
them all.
(10)
2.
(a) Let = (134)(254)(45) S5 . Write as the product of disjoint cycl
Math 120A Dis 10
Quiz 7
Fall 2011
Solution
1) (3 pts.) State the Fundamental Theorem of Finitely Generated Abelian Groups.
Every finitely generated abelian group G is isomorphic to a direct product of cyclic
groups of the form
pr pr m
1
1
n
n
Where the p
Math 120A Dis 10
Quiz 5
Fall 2011
Solution
1) (3 pts.) Let H be a subgroup of a group G , and let a G . Define the left coset of
H containing a and the right coset of H containing a .
The left coset of H containing a is the set aH ah h H .
The right coset
H OMEWORK 2
M ATH 120A (44700) FALL 2016
1. Canvas question. Let G be a group. Prove or disprove: We have (g1 g2 ) g3 = (g3 g2 ) g1
for all g1 , g2 , g3 2 G if and only if G is abelian.
2. This problem refers to units, which were defined in Exercise 1 of
H OMEWORK 3
M ATH 120A (44700) FALL 2016
1. Canvas question. Carefully prove that (Q\cfw_0 , ) is not isomorphic to (Q, +). (One possible
proof involves noticing that every rational number a can be written as b + b for some rational
number b. Another poss
H OMEWORK 1
M ATH 120A (44700) FALL 2016
Due Thursday, September 29th, in discussion section.
1. Canvas question. Answer both parts of this first question directly in Canvas. The goal of
this question is to practice your proof writing. If you submit an an
MATH 120A / Fall 2007 / HW8 solutions
11/30/07
SECTION 14
Problem 26. Assume G is abelian and T is the torsion subgroup of G (i.e. T is all the
elements of nite order). Show T is a normal subgroup of G and that G/T is torsion-free
(i.e. has no elements of
I NTRO G ROUP T HEORY (M ATH 120 A)
Midterm (solutions)
Problem 1.
What is the order of a subgroup of S8 generated by the permutation =
12345678
?
35487621
Solution. The permutation can be represented as a product of disjoint cycles:
= (1348)(257)
The or
1
Section 14
Problem 4. Find the order of the factor group Z3 Z5 /(cfw_0 Z5 ).
Note that cfw_0 Z5 is the same as the subgroup generated by the element (0, 1). 0 has
order 1 in Z3 and 1 has order 5 in Z5 , so the element (0, 1) has order LCM (1, 5) = 5 in