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) Let H = (2, 3) Z Z. Then H = cfw_(2k,
Math 301 Abstract Algebra
C.K. Li
Notes on Chapters 3-4
Chapter 3 Finite groups and Subgroups
Denition Let G be a group.
The order of G, denoted by |G| is the cardinality (nite or innite) of G.
The order of an element g P G is the smallest positive intege
Math 307 Abstract Algebra
C.K. Li
Notes on Chapters 5-6
Chapter 6/10 Isomorphisms/Homomorphisms
Denition Let G1 , G2 be groups with binary operations and . A function : G1 G2 is a
group homomorphism if pa bq paq pbq for all a, b P G1 . If, in addition, is
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
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
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
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
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 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 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 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 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 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 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
C.K. Li
Notes on Chapters10-11
Chapter 10 Group Homomorphisms
Denition
Let pG1 , 1 q, pG2 , 2 q be groups. Then a function : G1 G2 is a group homomorphism if
pa 1 bq paq 2 pbq
for all a, b P G1 .
The kernel of is the set Kerpq ta
Math 307 Abstract Algebra
C.K. Li
Notes on Chapters 12-13
Chapter 12
Denition
A ring is a set with two binary operations: addition a ` b and multiplication ab such that
(R1) pR, `q is an Abelian group with identity 0, and inverse a for a P R.
(R2) pabqc a
Math 307 Abstract Algebra
Sample Final Examination with solution
You will need to answer 10 out of 12 questions in the nal. Nine questions will be chosen from
the following list.
1. Suppose that H is a proper subgroup of Z under addition and H contains 18
Math 307 Abstract Algebra
Homework 8
Sample solution
1. (a) 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 iso
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
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 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 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 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 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 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 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 1
Solution 1
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 relatively prime, then ab need not divide c.
Since a|c and b|c, there exist m, n Z such th
Math 307 Abstract Algebra
C.K. Li
Notes on Chapters 14-15
Chapter 14 Ideals and Factor Rings
Denition
A subring A of a ring R is a (two-sided) ideal if ar, ra P A for every r P R and every a P A.
[Left and right absorbing power!]
Theorem 12.1 A non-empty