Math 307 Abstract Algebra
Homework 7
Sample solution
1. (a) Let G = cfw_3a 6b 10c : a, b, c Z under multiplication. Show that G is isomorphic to
3 6 10 .
(b) Let H = cfw_3a 6b 12c : a, b, c Z under multiplication. Show that G is NOT isomorphic to
3 6 12 .
Derivation
Homework 3
Due September 25, 2013
Solve the following problems. Please remember to use complete sentences and good grammar.
1. (4 Points) Let x, y Z. Prove that if 3 6 |x and 3 6 |y, then 3|(x2 y 2 ).
If 3 6 |x then x = 3P + 1 or x = 3P + 2 for
Math 307 FINAL
Printed-copy due: 5pm on 05/02/2014
Name:
(1) For fixed positive integers n1 and n2 , let
A1 0
D= A=
Mn (R) : A1 Mn1 (R), A2 Mn2 (R), n = n1 + n2
0 A2
(a) Prove that D is a subring of Mn (R).
First we note that the set D contains the zero
Math 307 Midterm
Due: 11:59pm on 03/09/2014
Name:
For the following problems, let S = cfw_1, . . . , n and Sn the permutation group of S.
1. Let = (a0 a1 . . . am1 ) Sn , i.e.,
(ai ) = a(i+1) mod m .
(a) Use induction to show that
n (ai ) = a(i+n) mod m
f
Math 307 PS3
Due: 5pm on 02/21/2014
Name:
1. [10 points] Prove that
1
H :=
0
n
:nZ
1
is a cyclic subgroup of GL2 (R).
To prove that H is a cyclic subgroup of the general linear group, we must show that a single element
a H is a generator. That is, H =< a
Math 307 Problem Set 2
Due: 5pm on 02/07/2014
Name:
1. [5 points] Let G be a group and g G. For all positive integers n, show that (g 1 )n = (g n )1 .
Prove by induction:
Base Case, n = 1:
(g 1 )n = (g n )1
(g 1 )1 = (g 1 )1
Anything raised to the first p
Math 307 Problem Set 1
Due: 23 January 2014
1. Let n be a fixed positive integer greater than 1. If a mod n = a0 and
b mod n = b0 , prove that
(a) [5 points] (a + b) mod n = (a0 + b0 ) mod n.
Proof. By the definition of congruence, a mod n = a0 n|(a a0 )
Math 307 PS4
Due: 5pm on 03/24/2014 (please print)
Name:
(1) [5 points] Suppose that H and K are subgroups of G and there are elements a and b in G such that
aH bK. Prove that H K.
Take h H and k K.
For the k K, there exists an a such that a = bk.
From th
Math 307 PS5
Printed-copy due: 5pm on 03/31/2014
Name:
(1) [10 points] Let M2 (R) be the group of all real 22 matrices under addition. Prove that M2 (R)
= R4 ,
where R4 is considered as a group under vector addition.
To show that these two spaces are iso
Math 307 Abstract Algebra
Sample final examination questions with solutions
1. Suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40,
Determine H.
Solution. Since gcd(18, 30, 40) = 2, there exists an x, y, z Z such that 18x
Math 307 PS6
Printed-copy due: 5pm on 04/14/2014
Name:
and be a homomorphism from G
to G.
(1) Let be a homomorphism from G to G
(a) [5 points] Prove that := is a homomorphism from G to G.
Take two elements g, h G.
Then must be equal to (gh) = (g)(h)
By
Vanessa Gray
Math 307 Abstract Algebra
Homework 1
Due: Sept. 3, 5:00 p.m.
Solve the following problems. Each problem is 5 points.
1. Suppose that a|c and b|c. If a and b are relatively prime, show that ab|c.
Show, by example, that if a, b are not relative
Math 307 Abstract Algebra
Homework 2
Sample Solution
1. Prove that the set of all 2 2 matrices with entries from R and determinant 1 is a group
under matrix multiplication.
Let M be the set of all 2 2 matrices with real entries such that for A M , det(A)
Math 307 Abstract Algebra
Homework 9
Sample solution
1. (a) Give an example of a subset of a ring that is a subgroup under addition but not a subring.
(b) Give an example of a nite non-commutative ring.
Solution. (a) Example 1. Let R = C and S = cfw_ix :
Math 307 Abstract Algebra
Homework 8
Sample solution
1. Let G be the group of nonzero real numbers under multiplication. Suppose r is a positive
integer. Show that x xr is a homomoprhism. Determine the kernel, and determine r so
that the map is an isomorp
Math 307 Abstract Algebra
Homework 6
Sample solution
1. If r is a divisor of m and s is a divisor of n, nd a subgroup of Zm Zn that is isomorphic
to Zr Zs .
Solution. Let a = m/r, b = n/s. Then H = cfw_(pa, qb) : p, q Z is a subgroup as (0, 0) H
and (p1 a
Math 307 Abstract Algebra
Homework 5
Sample solution
1. (a) Let H = (1, 2) S3 . Write down all the left cosets of H in S3 , and also the right cosets
of H in S3 .
Solution (1, 3)H = (1, 2, 3)H = cfw_(1, 3), (1, 2, 3), (2, 3)H = (1, 3, 2)H = cfw_(2, 3), (1
Math 307 Abstract Algebra Homework 3
Sample Solution
1. Let H = cfw_a + bi : a, b R, ab 0. Prove or disprove that H is a subgroup of C under
addition.
Solution. Note that 1, i H, but 1 + (i) H. So, H is not a subgroup.
/
2. Let H be a non-trivial subgroup
MATH 307
C.K. LI
Sample Exam. I
1. Let S be the set of all injective functions f : N N. Under the operation of function
composition, show that the group axioms (G0), (G1) and (G2) hold, but (G3) fails.
Solution. (G0) Suppose f, g : N N are one-one. If f (
Math 307
C.K. Li
Notes on Chapters 0 - 2
Chapter 0 Preliminaries
I assume that you are familiar with the material in Chapter 0 (Math 214).
Notation:
N Z Q R C.
Known results:
Well-ording principle; division algorithm on Z; greatest common divisor gcdpa, b
Math 307 Abstract Algebra
C.K. Li
Notes on Chapters 89
Chapter 8 External Direct products
Denition Let pG1 , 1 q, pG2 , 2 q be groups. The external direct product is G1 G2 tpg1 , g2 q :
g1 P G1 , g2 P G2 u and using the entry-wise operations px1 , x2 q py
Math 307 Abstract Algebra
C.K. Li
Notes on Chapters 6 7
Chapter 6 Isomorphisms
Denition Let G1 , G2 be groups. A function : G1 G2 is a group isomorphism if it is bijective
and satises pabq paqpbq for all a, b P G1 . If, in addition, G1 G2 , then is a grou
Math 307 Abstract Algebra Homework 3
Sample Solution
1. Let G be a group and a G.
(1) Show that C (a) = cfw_g G : ag = ga is a subgroup of G.
(2) Show that Z (G) = aG C (a).
Solution. (1) Let a G. Clearly, e C (a) because ae = ea. If x, y C (a), then (xy
Math 307 Abstract Algebra
Homework 2
Sample Solution
1. Prove that the set of all 2 2 matrices with entries from R and determinant 1 is a group
under matrix multiplication.
Let M be the set of all 2 2 matrices with real entries such that for A M , det(A)
Math 307 Abstract Algebra
Homework 6
Sample solution
1. If r is a divisor of m and s is a divisor of n, nd a subgroup of Zm Zn that is isomorphic
to Zr Zs .
Solution. Let a = m/r, b = n/s, H = cfw_(pa, qb) : p, q Z, and : Zr Zs H dened by
(p, q ) = (pa, q
Math 307 Abstract Algebra
Homework 7
Sample solution
1. (a) Let G = cfw_3a 6b 10c : a, b, c Z under multiplication. Show that G is isomorphic to
3 6 10 .
(b) Let H = cfw_3a 6b 12c : a, b, c Z under multiplication. Show that G is NOT isomorphic to
3 6 12 .
Math 307 Abstract Algebra
Homework 12
Sample solution
1. (a) Write x3 + 6 Z7 [x] as a product of irreducible polynomials over Z7 .
(b) Write x3 + x2 + x + 1 Z2 [x] as a product of irreducible polynomials over Z2 .
Solution. (a) Let f (x) = x3 + 6, then f
Math 307 Abstract Algebra
Homework 11
Sample Solution
1. List all the polynomials of degree 2 in Z2 [x]. Which of these are equal as functions from Z2
to Z2 , i.e., p(x) = q (x) for x = 0, 1?
Solution. f1 (x) = x2 , f2 (x) = x2 + x, f3 (x) = x2 + 1, f4 (x
Math 307 Abstract Algebra
Homework 10
Sample solution
1. (a) Give an example to show that the characteristic of a subring of a ring R may be dierent
from that of R.
(b) Show that the characteristic of a subdomain of an integral domain D is the same as tha