Difference between revisions of "MAT4233"

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Modern Abstract Algebra (3-0) 3 Credit Hours
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==Modern Abstract Algebra==
  
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==Description==
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[https://catalog.utsa.edu/undergraduate/sciences/mathematics/#courseinventory MAT 4233. Modern Abstract Algebra]. (3-0) 3 Credit Hours.
  
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The objective of this course is to introduce the basic concepts of abstract algebra, look into the interaction of algebraic operations with foundational constructions, such as products of sets and quotient sets, and to develop rigorous skills needed for further study.
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Prerequisites: [[MAT3233]].
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The course will focus on groups and homomorphisms, as well as provide an introduction to other algebraic structures like rings and fields.  
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Basic properties and examples of semigroups, monoids, and groups, detailed study of permutation, dihedral, and congruence groups, cyclic groups, normal subgroups, quotient groups, homomorphism, isomorphism theorems, direct products of groups, rings and fields and their basic properties, ideals, polynomial rings. Generally offered: Spring. Differential Tuition: $150.
  
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==Evaluation==
 
  
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* Midterms and an optional final.
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==Text==
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Charles C. Pinter. [https://store.doverpublications.com/0486474178.html ''A Book of Abstract Algebra.''] (Reprint of 2nd. ed. originally published in 1990) Dover Publications, NY (2010). ISBN-10: 0-486-47417-8.
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==Topics==
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{| class="wikitable sortable"
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! Date !! Sections !! Topics !! Student Learning Outcomes
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|-     
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|
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1
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||
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1 & 2
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||
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Binary operations on sets.
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|-
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|
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2
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3 & 4
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Groups: Definition and elementary properties.
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|-
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|
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3
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5
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Subgroups.
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* Exam score is the best of final score and midterm average.
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|-
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4
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7 & 8
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Groups of Permutations. Permutations of finite sets.
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* Students will have access to several past exams for practice.
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|-
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5
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9 & 10
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Group isomorphisms and orders of elements.
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==Text==
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|-
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J. Gallian, [https://isidore.co/calibre/get/pdf/4975 ''Contemporary abstract algebra''] (8e) Houghton Mifflin
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6
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---
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Review. First midterm exam.
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==Topics List==
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|-
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{| class="wikitable sortable"
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|
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! Week !! Session !! Topics !! Chapter !! Prerequisite Skills !! Learning Outcomes !! Examples !! Exercises
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7
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11
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Cyclic groups.
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|1
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|| Introduction to groups
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8
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13
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Counting cosets.
 
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*Symmetries
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*Properties of composition
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|-
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*Definition of a group
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*Elementary proofs with groups:
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9
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**uniqueness of identity
 
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**uniqueness of inverses
 
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**cancellation
 
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**shortcuts to establishing group axioms
 
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*Foundational examples with Cayley tables
 
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||Sets and functions
 
 
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*Motivation for the concept of a group
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14 & 15
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*Learn the definition of a group
 
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*Learn basic automatic properties of groups (with proofs) for later use as shortcuts
 
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*Starting to build a catalog of examples of groups
 
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*Learn to construct and read Cayley tables
 
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*Planting seeds for later concepts of ''subgroup'', ''factor group'', ''homomorphism'', ''isomorphism'', ''ring''
 
 
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*'''Z''', '''Q''', '''Q'''*, '''Q'''+, '''R''', '''R'''*, '''R'''+, {-1, 1}
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Homomorphisms and quotient groups.
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*'''R'''^n, M(n,'''R''')
 
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*symmetric group S_n
 
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*'''Z'''_2 defined for now as {even,odd} ({solids,stripes})
 
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*correspondence of '''Z'''_2 with {-1, 1} and with S_2
 
 
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|2
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||Introduction to homomorphisms
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10
 
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*Homomorphisms of groups
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---
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*Isomorphisms and their inverses
 
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*Examples
 
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*Cayley's theorem
 
 
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Review. Second midterm exam.
 
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*Functions
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*Groups
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|-
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*Matrix multiplication
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|
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11
 
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*Learn the definitions of homomorphism and isomorphism
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16
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*Starting to build a catalog of examples of homomorphisms
 
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*Prove that homomorphisms preserve powers for later use.
 
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*General framework for thinking of groups as symmetries
 
 
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*'''R''' -> '''R''': x -> ax
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The fundamental theorem of homomorphism.
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*'''R'''^n -> '''R'''^n: v-> Av
 
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*M(n,'''R''') -> M(n,'''R'''): X-> AX
 
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*'''R'''* -> '''R'''*: x -> x^n
 
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*'''R''' -> '''R'''*: x -> a^x
 
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*inclusions
 
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*natural projection '''Z''' -> '''Z'''_2
 
 
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|3
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||Groups in Linear Algebra and Complex Variable
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12
 
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* Euclidean space as an additive group
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17
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* Null space and column space of a linear map
 
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* Invertible linear transformations and matrices
 
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* Determinant
 
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* Additive and multiplicative subgroups of complex numbers
 
 
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*GL(n,'''R'''), O(n,'''R'''), SL(n,'''R'''), SO(n,'''R''')
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Rings and ideals.
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*'''C''', '''C'''*, S^1, ''n''-th roots of unity
 
 
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|15
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13
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18
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Ring homomorphisms.
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|-
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14
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19 & 20
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Quotient rings and integral domains.
 
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*Catch up and review for final
 
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* Study days
 
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|}
 
  
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==See also==
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<!-- |- -->
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<!-- 15 -->
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<!-- 21 & 22 -->
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<!-- The integers. Factorization into primes. -->
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<!-- 14 -->
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<!-- 23 -->
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<!-- Elements of number theory. -->
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<!-- || -->
  
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* [https://catalog.utsa.edu/undergraduate/coursedescriptions/mat/ UTSA Undergraduate Mathematics Course Descriptions]
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|-
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15
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---
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Wrap-up and review. Student Study Day.
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|-
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Latest revision as of 10:49, 25 March 2023

Modern Abstract Algebra

MAT 4233. Modern Abstract Algebra. (3-0) 3 Credit Hours.

Prerequisites: MAT3233. Basic properties and examples of semigroups, monoids, and groups, detailed study of permutation, dihedral, and congruence groups, cyclic groups, normal subgroups, quotient groups, homomorphism, isomorphism theorems, direct products of groups, rings and fields and their basic properties, ideals, polynomial rings. Generally offered: Spring. Differential Tuition: $150.


Text

Charles C. Pinter. A Book of Abstract Algebra. (Reprint of 2nd. ed. originally published in 1990) Dover Publications, NY (2010). ISBN-10: 0-486-47417-8.

Topics

Date Sections Topics Student Learning Outcomes

1

1 & 2

Binary operations on sets.

2

3 & 4

Groups: Definition and elementary properties.

3

5

Subgroups.

4

7 & 8

Groups of Permutations. Permutations of finite sets.

5

9 & 10

Group isomorphisms and orders of elements.

6

---

Review. First midterm exam.

7

11

Cyclic groups.

8

13

Counting cosets.

9

14 & 15

Homomorphisms and quotient groups.

10

---

Review. Second midterm exam.

11

16

The fundamental theorem of homomorphism.

12

17

Rings and ideals.

13

18

Ring homomorphisms.

14

19 & 20

Quotient rings and integral domains.


15

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Wrap-up and review. Student Study Day.

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