YTU – Faculty of Chemical andMetallurgical Engineering

Answers Sheet

YTU – Faculty of Chemical andMetallurgical Engineering

Answers Sheet

NOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTNOTE CHARTYTU – Faculty of Chemical andMetallurgical Engineering

Answers Sheet

YTU – Faculty of Chemical andMetallurgical Engineering

Answers Sheet 1. q1. q 2. q 3. q3. q 4. q4. q 5. q 6.q6.q 7. q7. q 8. q8. q 9. q 10.q10.q Total Student Name and

Surname--

Student Number--

Section Mathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical EngineeringMathematical Engineering Exam DateExam DateExam DateExam Date 06 / 06 / 201306 / 06 / 201306 / 06 / 2013

Course Name AlgebraAlgebra Group NumberGroup

NumberGroup

NumberExam

DurationExam

DurationExam

Duration90

min.90

min.Examination

Room Examination

Room Examination

Room Examination

Room Examination

Room KMB-202-203-210KMB-202-203-210

Course Instructor Name and Surname

Vedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAPVedat ŞİAP SignatureSignatureSignatureSignature

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Student Disciplinary Regulations "and to make or attempt to make copies of exams to" the actual perpetrators are suspended from one or two semesters. (YÖK; 2547 Student Disciplinary Regulations, 9. Article)

Please show all work! Answers without supporting work will not be given credit.

1. (24 points) Complete the following definitions. a) Let H be a subgroup of the group G . For any a in G ,

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

is a left coset of H in G .

b) Let H be a subgroup of the group G . The index of H in G , denoted by G :H[ ] , is

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

c) Langrange’s Theorem says that if H is a subgroup of a finite group G then

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

d) Let H be a subgroup of a group G . Then H is a normal subgroup of G if

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

e) The fundamental theorem of homomorphism states that if φ :G→G is an epimorphism then

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring if

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 1

g) Let D be a commutative ring with unity. Then D is an integral domain if

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

h) Let F be a commutative ring with unity. Then F is an field if

Exam III MAT 416 Spring 2013Name:

Directions: Please show all work and be as neat as possible.1.26 points Complete the following definitions.

a) Let H be a subgroup of the group G. For any a in G,

aH = {ah | h 2 H}is a left coset of H in G.

b) Let H be a subgroup of the group G. The index of H in G, denoted by [G : H], isthe number of distinct left cosets of H in G.

c) Lagrange’s Theorem says that if H is a subgroup of a finite group G then|G|= |H|[G : H].

d) Let H be a subgroup of a group G. Then H is a normal subgroup of G ifaH = Ha for all a 2 H, or equivalently if aha�1 2 H for all a 2 G and h 2 H.

e) The fundamental theorem of homomorphisms states that if f : G ! G0 is an epimorphism thenG

kerf⇠= G0.

f) Let R be a set which has two binary operations of addition and multiplication. Then R is a ring ifR is an abelian group with respect to addition, R is closed with respect to an associative multiplication, andmultiplication is distributive over addition from both the left and the right.

g) Let D be a commutative ring with unity. Then D is an integral domain ifD has no zero divisors.

h) Let F be a commutative ring with unity. Then F is an field ifevery nonzero element of F has a multiplicative inverse (or is a unit).

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 2

2. (9 points) Let G be the symmetric group of order 3, and let H be the subgroup e,σ{ } . The multiplication table for G is

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.

b) List the distinct right cosets of H in G and their elements.

c) Is H a normal subgroup of G?

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

b) List the distinct right cosets of H in G and their elements.

c) Is H a normal subgroup of G?

Page 2

a) List the distinct left cosets of H in G and their elements.

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.H = {e,s},rH = {r,g},r2H = {r2,d}

b) List the distinct right cosets of H in G and their elements.H = {e,s},Hr = {r,d},Hr2 = {r2,g}

c) Is H a normal subgroup of G?No, as rH 6= Hr.

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

H = {e,r,r2},sH = {s,d,g}

b) List the distinct right cosets of H in G and their elements.H = {e,r,r2},sH = {s,g,d}

c) Is H a normal subgroup of G?Yes, since the left cosets are the same as the corresponding right cosets.

Page 2

b) List the distinct right cosets of H in G and their elements.

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.H = {e,s},rH = {r,g},r2H = {r2,d}

b) List the distinct right cosets of H in G and their elements.H = {e,s},Hr = {r,d},Hr2 = {r2,g}

c) Is H a normal subgroup of G?No, as rH 6= Hr.

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

H = {e,r,r2},sH = {s,d,g}

b) List the distinct right cosets of H in G and their elements.H = {e,r,r2},sH = {s,g,d}

c) Is H a normal subgroup of G?Yes, since the left cosets are the same as the corresponding right cosets.

Page 2

c) Is H a normal subgroup of G ?

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.H = {e,s},rH = {r,g},r2H = {r2,d}

b) List the distinct right cosets of H in G and their elements.H = {e,s},Hr = {r,d},Hr2 = {r2,g}

c) Is H a normal subgroup of G?No, as rH 6= Hr.

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

H = {e,r,r2},sH = {s,d,g}

b) List the distinct right cosets of H in G and their elements.H = {e,r,r2},sH = {s,g,d}

c) Is H a normal subgroup of G?Yes, since the left cosets are the same as the corresponding right cosets.

Page 2

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 3

3. (9 points)Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup e,ρ,ρ2{ } . a) List the distinct left cosets of H in G and their elements.

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.H = {e,s},rH = {r,g},r2H = {r2,d}

b) List the distinct right cosets of H in G and their elements.H = {e,s},Hr = {r,d},Hr2 = {r2,g}

c) Is H a normal subgroup of G?No, as rH 6= Hr.

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

H = {e,r,r2},sH = {s,d,g}

b) List the distinct right cosets of H in G and their elements.H = {e,r,r2},sH = {s,g,d}

c) Is H a normal subgroup of G?Yes, since the left cosets are the same as the corresponding right cosets.

Page 2

b) List the distinct right cosets of H in G and their elements.

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.H = {e,s},rH = {r,g},r2H = {r2,d}

b) List the distinct right cosets of H in G and their elements.H = {e,s},Hr = {r,d},Hr2 = {r2,g}

c) Is H a normal subgroup of G?No, as rH 6= Hr.

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

H = {e,r,r2},sH = {s,d,g}

b) List the distinct right cosets of H in G and their elements.H = {e,r,r2},sH = {s,g,d}

c) Is H a normal subgroup of G?Yes, since the left cosets are the same as the corresponding right cosets.

Page 2

c) Is H a normal subgroup of G ?

Exam III MAT 416 Spring 2013

2.9 points Let G be the symmetric group of order 3, and let H be the subgroup {e,s}. The multiplication table for G is

· e r r2 s g de e r r2 s g dr r r2 e g d sr2 r2 e r d s gs s d g e r2 rg g s d r e r2d d g s r2 r e

a) List the distinct left cosets of H in G and their elements.H = {e,s},rH = {r,g},r2H = {r2,d}

b) List the distinct right cosets of H in G and their elements.H = {e,s},Hr = {r,d},Hr2 = {r2,g}

c) Is H a normal subgroup of G?No, as rH 6= Hr.

3.9 points Let G be the symmetric group of order 3 as in number 2. Let H be the subgroup {e,r,r2}.a) List the distinct left cosets of H in G and their elements.

H = {e,r,r2},sH = {s,d,g}

b) List the distinct right cosets of H in G and their elements.H = {e,r,r2},sH = {s,g,d}

c) Is H a normal subgroup of G?Yes, since the left cosets are the same as the corresponding right cosets.

Page 2

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 4

4. (10 points)Let H and K be a normal subgroups of a group G . Prove that H ∩K is a normal subgroup of G . You may assume that H ∩K is a subgroup of G .

Exam III MAT 416 Spring 2013

4.9 points Let H and K be a normal subgroups of a group G. Prove that H\K is a normal subgroup of G. You may assumethat H \K is a subgroup of G.

Proof. Let h 2 H \K and x 2 G. Then h 2 H and so xhx�1 2 H as H is a normal subgroup of G. Also h 2 Kand so xhx�1 2 K as K is a normal subgroup of G. Thus xhx�1 2 H\K. Therefore H\K is a normal subgroupof G.

5.9 points Let G be the multiplicative group consisting of all invertible elements of Z26 under multiplication.So G = {[1], [3], [5], [7], [9], [11], [15], [17], [19], [21], [23], [25]}. Let H = {[1], [3], [9]}. List the distinct elementsof G/H and make a multiplication table for G/H.

G/H = {H, [5]H, [7]H, [17]H}

· H [5]H [7]H [17]HH H [5]H [7]H [17]H

[5]H [5]H [17]H H [7]H[7]H [7]H H [17]H [5]H[17]H [17]H [7]H [5]H H

6.10 points Consider the additive groups Z and Z8 and the epimorphism f : Z! Z8 defined by f(x) = [x].a) Find the elements of K = kerf.

K = {8m | m 2 Z}

b) List the distinct elements of Z/K.Z/K = {K,1+K,2+K,3+K,4+K,5+K,6+K,7+K}

Page 3

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 5

5. (10 points)Let G be the multiplicative group consisting of all invertible elements of 26 under multiplication. Let H = 1[ ], 3[ ], 9[ ]{ } . List the distinct elements of G H and make a multiplication table for G H .

Exam III MAT 416 Spring 2013

4.9 points Let H and K be a normal subgroups of a group G. Prove that H\K is a normal subgroup of G. You may assumethat H \K is a subgroup of G.

Proof. Let h 2 H \K and x 2 G. Then h 2 H and so xhx�1 2 H as H is a normal subgroup of G. Also h 2 Kand so xhx�1 2 K as K is a normal subgroup of G. Thus xhx�1 2 H\K. Therefore H\K is a normal subgroupof G.

5.9 points Let G be the multiplicative group consisting of all invertible elements of Z26 under multiplication.So G = {[1], [3], [5], [7], [9], [11], [15], [17], [19], [21], [23], [25]}. Let H = {[1], [3], [9]}. List the distinct elementsof G/H and make a multiplication table for G/H.

G/H = {H, [5]H, [7]H, [17]H}

· H [5]H [7]H [17]HH H [5]H [7]H [17]H

[5]H [5]H [17]H H [7]H[7]H [7]H H [17]H [5]H[17]H [17]H [7]H [5]H H

6.10 points Consider the additive groups Z and Z8 and the epimorphism f : Z! Z8 defined by f(x) = [x].a) Find the elements of K = kerf.

K = {8m | m 2 Z}

b) List the distinct elements of Z/K.Z/K = {K,1+K,2+K,3+K,4+K,5+K,6+K,7+K}

Page 3

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 6

6. (10 points)Consider the additive groups and 8 and the epimorphism φ :→ 8 defined by φ x( ) = x[ ].a) Find the elements of K = kerφ .

Exam III MAT 416 Spring 2013

4.9 points Let H and K be a normal subgroups of a group G. Prove that H\K is a normal subgroup of G. You may assumethat H \K is a subgroup of G.

Proof. Let h 2 H \K and x 2 G. Then h 2 H and so xhx�1 2 H as H is a normal subgroup of G. Also h 2 Kand so xhx�1 2 K as K is a normal subgroup of G. Thus xhx�1 2 H\K. Therefore H\K is a normal subgroupof G.

5.9 points Let G be the multiplicative group consisting of all invertible elements of Z26 under multiplication.So G = {[1], [3], [5], [7], [9], [11], [15], [17], [19], [21], [23], [25]}. Let H = {[1], [3], [9]}. List the distinct elementsof G/H and make a multiplication table for G/H.

G/H = {H, [5]H, [7]H, [17]H}

· H [5]H [7]H [17]HH H [5]H [7]H [17]H

[5]H [5]H [17]H H [7]H[7]H [7]H H [17]H [5]H[17]H [17]H [7]H [5]H H

6.10 points Consider the additive groups Z and Z8 and the epimorphism f : Z! Z8 defined by f(x) = [x].a) Find the elements of K = kerf.

K = {8m | m 2 Z}

b) List the distinct elements of Z/K.Z/K = {K,1+K,2+K,3+K,4+K,5+K,6+K,7+K}

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b) List the distinct elements of K .

Exam III MAT 416 Spring 2013

4.9 points Let H and K be a normal subgroups of a group G. Prove that H\K is a normal subgroup of G. You may assumethat H \K is a subgroup of G.

Proof. Let h 2 H \K and x 2 G. Then h 2 H and so xhx�1 2 H as H is a normal subgroup of G. Also h 2 Kand so xhx�1 2 K as K is a normal subgroup of G. Thus xhx�1 2 H\K. Therefore H\K is a normal subgroupof G.

5.9 points Let G be the multiplicative group consisting of all invertible elements of Z26 under multiplication.So G = {[1], [3], [5], [7], [9], [11], [15], [17], [19], [21], [23], [25]}. Let H = {[1], [3], [9]}. List the distinct elementsof G/H and make a multiplication table for G/H.

G/H = {H, [5]H, [7]H, [17]H}

· H [5]H [7]H [17]HH H [5]H [7]H [17]H

[5]H [5]H [17]H H [7]H[7]H [7]H H [17]H [5]H[17]H [17]H [7]H [5]H H

6.10 points Consider the additive groups Z and Z8 and the epimorphism f : Z! Z8 defined by f(x) = [x].a) Find the elements of K = kerf.

K = {8m | m 2 Z}

b) List the distinct elements of Z/K.Z/K = {K,1+K,2+K,3+K,4+K,5+K,6+K,7+K}

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! ALGEBRA! 7

7. (10 points)

a) List all zero divisors in 18.Exam III MAT 416 Spring 2013

7.10 points a) List all zero divisors in Z18.[2], [3], [4], [6], [8], [9], [10], [12], [14], [15], [16]

b) List all units in Z18.[1], [5], [7], [11], [13], [17]

8.9 points Let D be an integral domain. Prove that if x 2 D satisfies x2 = x then either x = 0 or x = 1.

Proof. Assume x 2 D satisfies x2 = x. Thenx2 = x

=) x2 � x = 0 subtracting x from both sides=) x(x�1) = 0 using the distributive property

Since D is an integral domain, D contains no zero divisors. Hence this implies that either x = 0 or x� 1 = 0.In other words either x = 0 or x = 1.

9.9 points Prove that the set of all real numbers of the form a+bp

17 with a,b 2 Z is a subring of the ring of real numbers.

Proof. Let S = {m+np

17 | m,n 2 Z}. First observe that 0 = 0+0p

17 2 S so that S 6= /0.Now let m+n

p17, p+q

p17 2 S for some m,n, p,q 2 Z. Then

(m+np

17)� (p+qp

17) = (m� p)+(n�q)p

17 2 S since m� p,n�q 2 Zand

(m+np

17)(p+qp

17) = (mp+17nq)+(mq+np)p

17 2 S since mp+17nq,mq+np 2 Z.Therefore S is a subring of R by Theorem 5.4.

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b) List all units in 18.

Exam III MAT 416 Spring 2013

7.10 points a) List all zero divisors in Z18.[2], [3], [4], [6], [8], [9], [10], [12], [14], [15], [16]

b) List all units in Z18.[1], [5], [7], [11], [13], [17]

8.9 points Let D be an integral domain. Prove that if x 2 D satisfies x2 = x then either x = 0 or x = 1.

Proof. Assume x 2 D satisfies x2 = x. Thenx2 = x

=) x2 � x = 0 subtracting x from both sides=) x(x�1) = 0 using the distributive property

Since D is an integral domain, D contains no zero divisors. Hence this implies that either x = 0 or x� 1 = 0.In other words either x = 0 or x = 1.

9.9 points Prove that the set of all real numbers of the form a+bp

17 with a,b 2 Z is a subring of the ring of real numbers.

Proof. Let S = {m+np

17 | m,n 2 Z}. First observe that 0 = 0+0p

17 2 S so that S 6= /0.Now let m+n

p17, p+q

p17 2 S for some m,n, p,q 2 Z. Then

(m+np

17)� (p+qp

17) = (m� p)+(n�q)p

17 2 S since m� p,n�q 2 Zand

(m+np

17)(p+qp

17) = (mp+17nq)+(mq+np)p

17 2 S since mp+17nq,mq+np 2 Z.Therefore S is a subring of R by Theorem 5.4.

Page 4

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! ALGEBRA! 8

8. (9 points) Let D be an integral domain. Prove that if x ∈D satisfies x2 = x then either x = 0 or x = 1 .

Exam III MAT 416 Spring 2013

7.10 points a) List all zero divisors in Z18.[2], [3], [4], [6], [8], [9], [10], [12], [14], [15], [16]

b) List all units in Z18.[1], [5], [7], [11], [13], [17]

8.9 points Let D be an integral domain. Prove that if x 2 D satisfies x2 = x then either x = 0 or x = 1.

Proof. Assume x 2 D satisfies x2 = x. Thenx2 = x

=) x2 � x = 0 subtracting x from both sides=) x(x�1) = 0 using the distributive property

Since D is an integral domain, D contains no zero divisors. Hence this implies that either x = 0 or x� 1 = 0.In other words either x = 0 or x = 1.

9.9 points Prove that the set of all real numbers of the form a+bp

17 with a,b 2 Z is a subring of the ring of real numbers.

Proof. Let S = {m+np

17 | m,n 2 Z}. First observe that 0 = 0+0p

17 2 S so that S 6= /0.Now let m+n

p17, p+q

p17 2 S for some m,n, p,q 2 Z. Then

(m+np

17)� (p+qp

17) = (m� p)+(n�q)p

17 2 S since m� p,n�q 2 Zand

(m+np

17)(p+qp

17) = (mp+17nq)+(mq+np)p

17 2 S since mp+17nq,mq+np 2 Z.Therefore S is a subring of R by Theorem 5.4.

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Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 9

9. (9 points)Prove that the set of all real numbers of the form a + b 13 with a,b∈ is a ring.

Exam III MAT 416 Spring 2013

7.10 points a) List all zero divisors in Z18.[2], [3], [4], [6], [8], [9], [10], [12], [14], [15], [16]

b) List all units in Z18.[1], [5], [7], [11], [13], [17]

8.9 points Let D be an integral domain. Prove that if x 2 D satisfies x2 = x then either x = 0 or x = 1.

Proof. Assume x 2 D satisfies x2 = x. Thenx2 = x

=) x2 � x = 0 subtracting x from both sides=) x(x�1) = 0 using the distributive property

Since D is an integral domain, D contains no zero divisors. Hence this implies that either x = 0 or x� 1 = 0.In other words either x = 0 or x = 1.

9.9 points Prove that the set of all real numbers of the form a+bp

17 with a,b 2 Z is a subring of the ring of real numbers.

Proof. Let S = {m+np

17 | m,n 2 Z}. First observe that 0 = 0+0p

17 2 S so that S 6= /0.Now let m+n

p17, p+q

p17 2 S for some m,n, p,q 2 Z. Then

(m+np

17)� (p+qp

17) = (m� p)+(n�q)p

17 2 S since m� p,n�q 2 Zand

(m+np

17)(p+qp

17) = (mp+17nq)+(mq+np)p

17 2 S since mp+17nq,mq+np 2 Z.Therefore S is a subring of R by Theorem 5.4.

Page 4

Please show all work! Answers without supporting work will not be given credit.

! ALGEBRA! 10