Let H be a normal subgroup of a group G. Define j: G ® G/H by: (a) j(ab) = j(a)j(b) (b) j(a) = Ha (c) j(Ha) = a (d) j(ab) = Hab (e) j(ab) = HaHb.

Slides:

Advertisements

Similar presentations

Is It A Prime Number, Yes or No? Jonathan Brinkerhoff.

Advertisements

Quiz April Sections 1.6,1.7,1.8,4.1,4.2. Quiz: Apr. 20 ’04: pm 1.Set S1 is the set of all Irish citizens with blue eyes. Set S2 is the.

 Definition 16: Let H be a subgroup of a group G, and let a  G. We define the left coset of H in G containing g,written gH, by gH ={g*h| h  H}. Similarity.

Section 13 Homomorphisms Definition A map  of a group G into a group G’ is a homomorphism if the homomophism property  (ab) =  (a)  (b) Holds for.

If H is the subgroup in Z 12, then H2 = (a) 5(b) 14 (c) {3, 6, 9, 0}  {2}(d) {5, 8, 11, 2} (e) {5, 6, 9, 0}(f) {3, 2, 5}

I. Homomorphisms & Isomorphisms II. Computing Linear Maps III. Matrix Operations VI. Change of Basis V. Projection Topics: Line of Best Fit Geometry of.

How do we start this proof? (a) Assume A n is a subgroup of S n. (b)  (c) Assume o(S n ) = n! (d) Nonempty:

What is the first line of the proof? a). Assume G has an Eulerian circuit. b). Assume every vertex has even degree. c). Let v be any vertex in G. d). Let.

Quiz April Sections 1.6,1.7,1.8. Quiz: Apr. 21 ’05: pm 1. Consider the following sets: Provide the following sets using set-builder.

What is the first line of the proof? 1.If a divides b, then a divides b – c. 2.If a divides b, then a divides c. 3.Assume a divides b – c. 4.Assume a divides.

How do we start this proof? (a) Let x  gHg -1. (b)  (c) Assume a, b  H (d) Show Nonempty:

Let G be a group. Define  : G ® G by  ( x ) = x. The First Isomorphism Theorem says: (a) j(ab) = j(a)j(b)(b) j is onto. (c) G  G(d) G  {e} (e) G/{e}

Section 6.2 One-to-One Functions; Inverse Functions.

What is the best way to start? 1.Plug in n = 1. 2.Factor 6n 2 + 5n Let n be an integer. 4.Let n be an odd integer. 5.Let 6n 2 + 5n + 4 be an odd.

Section 5.2 One-to-One Functions; Inverse Functions.

X’morphisms & Projective Geometric J. Liu. Outline  Homomorphisms 1.Coset 2.Normal subgrups 3.Factor groups 4.Canonical homomorphisms  Isomomorphisms.

A Very Practical Series 1 What if we also save a fixed amount (d) every year?

Let A = {1,2,3,4} and B = {a,b,c}. Define the relation f from A to B by f = {(1,b), (2,a), (3,c), (4,b)}. Is f a function? (1) Yes (2) No.

Section 4.1 Finite Permutation Groups Permutation of a Set Let A be the set { 1, 2, …, n }. A permutation on A is a function f : A  A that is both one-to-one.

(a) (b) (c) (d). What is (1,2,3)  (3,4,2)? (a) (1, 2, 3, 4) (b) (1,2)  (3,4) (c) (1,3,4,2) (d) (3,1)  (4,2)

1.1 Key Concepts.

Ms. Battaglia AP Calculus. The inverse function of the natural logarithmic function f(x)=lnx is called the natural exponential function and is denoted.

Lesson 1-4 Example Example 1 Graph the even whole numbers between 9 and Locate 9 and 19 on the number line. Circle them to remind yourself.

Divisibility Rules and Mental Math

ⅠIntroduction to Set Theory 1. Sets and Subsets

4x + 2y = 18 a. (1,8) = 18 a. 4(1) + 2(8) = 18 b. (3,3)20 = 18 b. 4(3) + 2(3) = = = 18 15x + 5y = 5 a. (-2,7) NoYes b. (-1,4) a.

1.5 Divisibility Rules SRB pg 11..

Name points, lines, and planes Chapter 1.1 Undefined Terms Point Line Plane Space Defined Terms Segment Ray What are collinear and coplanar points? What.

 A -  B -  C -  D - Yes No Not sure.  A -  B -  C -  D - Yes No Not sure.

Great Theoretical Ideas in Computer Science for Some.

Math 71B 9.2 – Composite and Inverse Functions 1.

Math 3121 Abstract Algebra I Lecture 5 Finish Sections 6 + Review: Cyclic Groups, Review.

Math 3121 Abstract Algebra I

Math 3121 Abstract Algebra I Lecture 11 Finish Section 13 Section 14.

Section 14 Factor Groups Factor Groups from Homomorphisms. Theorem Let  : G  G’ be a group homomorphism with kernel H. Then the cosets of H form a factor.

CS Lecture 14 Powerful Tools     !. Build your toolbox of abstract structures and concepts. Know the capacities and limits of each tool.

A.4a Polynomial Division Keep away from people who try to belittle your ambitions. Small people always do that, but the really great make you feel that.

Math 3121 Abstract Algebra I Lecture 14 Sections

Chapter 3: Functions and Graphs 3.6: Inverse Functions Essential Question: How do we algebraically determine the inverse of a function.

 Theorem 6.21: Let H be a subgroup of G. H is a normal subgroup of G iff g -1 hg  H for  g  G and h  H.  Proof: (1) H is a normal subgroup of G.

1.4 Solving Absolute Value Equations Evaluate and solve Absolute Value problems.

I-Geometry Semester 1 Quiz 9 Prize Show

S3 Credit Mathematics Similar Triangles There now follows a short test.

Hello. ok Hello ja.

Math 3121 Abstract Algebra I Lecture 15 Sections

Garis-garis Besar Perkuliahan

Group A set G is called a group if it satisfies the following axioms. G 1 G is closed under a binary operation. G 2 The operation is associative. G 3 There.

Isomorphisms and Isomorphic Groups (10/9) We can now say what we mean by two groups being “the same” even though their operations and elements may look.

Inverse Functions. DEFINITION Two relations are inverses if and only if when one relation contains (a,b), the other relation contains (b,a).

CS336 F07 Counting 2. Example Consider integers in the set {1, 2, 3, …, 1000}. How many are divisible by either 4 or 10?

Lesson 1a Notes – Simplifying Square Root Expressions

Abstract Algebra I.

Math 3121 Abstract Algebra I

Groups and Applications

Math 3121 Abstract Algebra I

Math 3121 Abstract Algebra I

Homomorphisms (11/20) Definition. If G and G’ are groups, a function  from G to G’ is called a homomorphism if it is operation preserving, i.e., for all.

75 Has 7 in the tens place Odd Number Less than 100

Let H be a normal subgroup of G, and let G/H={Hg|gG}

Objectives Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems.

6.4.3 Lagrange's Theorem Theorem 6.19: Let H be a subgroup of the group G. Then {gH|gG} and {Hg|gG} have the same cardinal number Proof：Let S={Hg|gG}

Definition 19: Let [H;. ] be a normal subgroup of the group [G;. ]

For g2G2, (n)(N), g1G1 s.t (g1)=g2 ? surjection homomorphism.

2. From Groups to Surfaces

Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28

12 Has 1 in the tens place Even Number Less than 20

Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28

Multiplying integers 3(3) = (3) = (3) = (3) = 0 -3

EXAMPLE 2 Name segments, rays, and opposite rays

Presentation transcript:

Let H be a normal subgroup of a group G. Define j: G ® G/H by: (a) j(ab) = j(a)j(b) (b) j(a) = Ha (c) j(Ha) = a (d) j(ab) = Hab (e) j(ab) = HaHb

Let H be a normal subgroup of a group G. Define j: G ® G/H by j(a) = Ha. Is j a homomorphism? (a) Yes (b) No

Let H be a normal subgroup of a group G. Define j: G ® G/H by j(a) = Ha. Is j one-to-one? (a) Yes (b) No

Let H be a normal subgroup of a group G. Define j: G ® G/H by j(a) = Ha. Is j onto? (a) Yes (b) No

Let G be any group. Define j: G ® G  Z 2 by: (a) j(ab) = j(a)j(b) (b) j(g) = (g 1, g 2 ) (c) j(g 1, g 2 ) = g 1 (d) j(a) = (a, 0) (e) j(g) = (g, 1)

Let G be any group. Define j: G ® G  Z 2 by j(a) = (a, 0). Is j a homomorphism? (a)Yes (b) No

Let G be any group. Define j: G ® G  Z 2 by j(a) = (a, 0). Is j one-to-one? (a)Yes (b) No

Let G be any group. Define j: G ® G  Z 2 by j(a) = (a, 0). Is j onto? (a)Yes (b) No

Define j: S n ® Z 2 by: (a) j(ab) = j(a)j(b) (b) j(a) = Ha (c) j(g 1, g 2 ) = g 1 g 2 (d) j(f) = 0 if f is even, 1 if f is odd (e) j(f) = 0 if f is odd, 1 if f is even

Define j: S n ® Z 2 by j(f) = 0 if f is even, 1 if f is odd Is j a homomorphism? Yes No

Define j: S n ® Z 2 by j(f) = 0 if f is even, 1 if f is odd Is j one-to-one? Yes No

Define j: S n ® Z 2 by j(f) = 0 if f is even, 1 if f is odd Is j onto? Yes No