# MA231: Abstract Algebra I

Unit 2: Fundamentals of Groups   Groups are the most fundamental of all algebraic structures.  Consisting of a set with an operator (often called a binary operator), groups are simple, yet powerful, entities with applications in fields such as physics and economics.  With fewer properties than, say, the set of real numbers with addition and multiplication, general groups give rise to structures that are as elegant as they are sometimes strange.  As such, the study of groups is often where students of mathematics “trip up.”  To avoid problems, do not make assumptions that are not available.  For instance, we are aware that the field of real numbers is commutative under the property of multiplication.  That is, for any real numbers a and b, a*b = b*a.  However, while n?n matrices are groups under matrix multiplication, we can show examples where AB ? BA.

We will then study some examples of groups, beginning with the *
finite group, which you will likely find easy to study.  We will also look at some special types of groups, such as cyclic and permutation groups.  Cyclic groups are generated from a single element.  In fact, if the set G contained nothing but powers of some element g, then G = <g> = {gn |n is an integer} and g would be called the “generator of G.”  Interestingly, a certain set of permutations of some set M is also a group (a **permutation group).  The set of all possible permutations on M is called the symmetric group of M. This is an important group that has relevance in Abstract Algebra II, when we will study Galois Theory.  Another important example, especially in Linear Algebra, is the general linear group of invertible matrices.  You will see more of this group in Abstract Algebra II.*

After defining and examining a few examples of groups, we will see that any subset of a set with group properties for an operator is itself a group if it has the same properties.  These groups are called “subgroups.”  We will also consider functions from one group to another.  Any function (also called a “mapping”) that retains consistent properties from one group to the next is called a homomorphism.  That is, as we consider an operation * on group G and ? on group H, if some mapping f on G paired elements of G with those of H (that is, f(g) = h for some g in G and h in H) such that f(a*b) = f(a)?f(b), then f would be a *homomorphism** of G into H, and G and H would be considered homomorphic.  If the mapping f were also 1-1 and onto (that is, the domain or image of the mapping covers all of H), f would be an isomorphism.  The study of such mappings is important, for if H were a simpler group than G and we discovered that H and G were isomorphic, then anything we discover about H is also true of G.*

We will end this unit with cosets.  Cosets are formed of true subgroups of one group and single elements of the larger group.  If all cosets are of the form gH = Hg, where H is a subgroup of G and g is in G, then H is called *normal.  This is important to remember because, in general, gH may **not be equal to Hg.  This is why we cannot assume commutativity.  Cosets are analogous to equivalence classes on the integers, because they partition groups into distinct sets.  Factor groups are normal subgroups.*

This unit will take you 39 hours to complete.

☐    Subunit 2.1: 4 hours

☐    Subunit 2.2: 12 hours

☐    Subunit 2.2.1: 3 hours

☐    Subunit 2.2.2: 3 hours

☐    Subunit 2.2.3: 3 hours

☐    Subunit 2.2.4: 1.5 hours

☐    Subunit 2.2.5: 1.5 hours

☐    Subunit 2.3: 4 hours

☐    Subunit 2.4: 5 hours

☐    Subunit 2.5: 4 hours

☐    Subunit 2.6: 10 hours

☐    Subunit 2.6.1: 5 hours

☐    Subunit 2.6.2: 5 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define and generate groups.
• Determine whether or not a group is cyclic.
• Determine whether or not a mapping is a homomorphism or isomorphism.
• Find all the cosets of a particular set.

2.1 Definition of a Group   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups” (PDF)

Also available in:

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Instructions: For the section on examples of sets, read “3.2
Definitions and Examples,” pages 40-47.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be found
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• Lecture: YouTube: Nathan Carter's “Visual Group Theory”: “Part I” and “Visual Group Theory”: “Part II” Link: YouTube: Nathan Carter's “Visual Group Theory”: “Part I” (YouTube) and “Visual Group Theory”: “Part II” (YouTube)

Instructions: Please click on the links above, and view each video in its entirety (1:28 minutes for “Part I” and 9:58 minutes for “Part II”).  These video lecture is an interesting way of discussing groups in terms of visual symmetries.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 2, 8, 15, and 17” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 2, 8, 15, and 17” (PDF)

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Instructions: Work through problems 2, 8, 15, and 17 on page 51.  After you complete each exercise, please see the solutions on page

1.

2.2 Examples of Groups   2.2.1 Finite Groups   - Reading: Wolfram MathWorld: Eric W. Weisstein’s “Finite Group” Link: Wolfram MathWorld: Eric W. Weisstein’s “Finite Group” (HTML)

Instructions: Please click on the link to read the information on the webpage.  This webpage contains a concise rendering of information on finite groups.  It also contains a visual representation of several common finite groups.

Instructions: Please read the entire web page.  This webpage contains a concise rendering of information on finite groups.  It also contains a visual representation of several common finite groups.

• Web Media: YouTube: Klein Four's “Finite Simple Group (of Order 2)” Humor Video Link: YouTube: Klein Four’s “Finite Simple Group (of Order 2)” (YouTube) Humor Video

Instructions: Please note that viewing this video is optional and this humorous video is meant to entertain.  Who says mathematicians can't laugh?  This is a now-famous piece of group theory humor set to a four-part a capella arrangement.  Klein Four, a group of Northwestern University mathematicians, recorded this in front of an audience in their math department in November, 2006.  Though intended strictly for humor, the use of terminology is correct and useful information can still be gleaned from the video.  Please click on the link, and view the video in its entirety (about 3 minutes).

2.2.2 Cyclic Groups   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups” (PDF)

Also available in:

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Instructions: For the section on examples of sets, read Chapter 4,
“Cyclic Groups”, pages 57-73.

use displayed on pages 410 417 of the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be found
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Instructions: Click on the link above, and read the webpage in its entirety for a discussion on cyclic groups and good examples of cyclic groups.  The page contains information on properties of cyclic groups and ties cyclic groups to other group types that will be covered later.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups”: “Exercise Problems 3 and 4” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cyclic Groups”: “Exercise Problems 3 and 4” (PDF)

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Instructions: Complete problems 3 and 4 on page 69.  After you have finished each problem, check the solutions on page 397.

2.2.3 Permutation Groups   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups” (PDF)

Also available in:

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Instructions: For the section on examples of sets, read Chapter 5
“Permutation Groups”, pages 74-91.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be found
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Instructions: Please click on the link, and view the video in its entirety.  The video discusses a permutation group of a regular geometric figure (a Star of David).  This is an interesting visual presentation.

Terms of Use: The linked material above has been reposted by the kind permission of Youtube User: S22105 and can be viewed in its original form here.  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups”: “Exercise Problems 1, 2, and 3” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Permutation Groups”: “Exercise Problems 1, 2, and 3” (PDF)

Also available in:

Instructions: Work through problems 1, 2, and 3 on page 88.  After you complete these, check your answers on page 398.

2.2.4 Symmetric Group   - Reading: Wolfram MathWorld: Eric. W. Weisstein’s “Symmetric Group” Link: Wolfram MathWorld: Eric. W. Weisstein’s “Symmetric Group”  (HTML)

Instructions: Please click on the link to read the information on the webpage.  This webpage contains a concise rendering of information on symmetric groups.  It also contains a visual representation of symmetric group multiplication table.

Instructions: Please read the entire webpage.  This webpage contains a concise rendering of information on finite groups.  It also contains a visual representation of several common finite groups.

2.2.5 General Linear Group of Invertible Matrices   - Reading: Wolfram MathWorld: Eric W. Weisstein’s “General Linear Group” Link: Wolfram MathWorld: Eric W. Weisstein’s “General Linear Group” (HTML)

Instructions: Please click on the link to read the information on the webpage for a concise rendering of information on the general linear group.  It is short but has a number of links to related topics of interest.

• Reading: Wikipedia: “General Linear Group” Link: Wikipedia: “General Linear Group” (PDF)

Instructions: Please read the entire webpage for a concise rendering of information on general linear groups, including various examples.  This topic is useful toward Abstract Algebra II.

2.3 Subgroups   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups” (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “3.3
Subgroups,” pages 47-50.

use displayed on pages 410-417 on the PDF file.  The material linked
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attributed to Thomas W. Judson and the original version can be found
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Instructions: Please click on the link, and view the video in its entirety (5:42 minutes) to learn about subgroups.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Groups”: “Exercise Problems 34 and 40” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications“Groups”: “Exercise Problems 34 and 40” (PDF)

Also available in:

Instructions: Complete problems 34 and 40 on page 52.  Then, check your answers against the solutions on page 397.

2.4 Homomorphisms   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms” (PDF)

Also available in:

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Instructions: For the section on examples of sets, read “Chapter
11: Homomorphisms,” pages 165-171.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be found
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Instructions: Please click on the link, and view the video in its entirety (7:39 minutes) for a discussion on group homomorphisms and isomorphisms in a fairly straight-forward manner.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms”: “Exercise Problems 2, 4, and 9” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Homomorphisms”: “Exercise Problems 2, 4, and 9” (PDF)

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Instructions: Try to do problems 2, 4, and 9 on page 173.  After you complete the assigned problems, check the solutions on page

1.

2.5 Isomorphisms   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms” (PDF)

Also available in:

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Instructions: For the section on examples of sets, read “Chapter 9:
Isomorphisms,” pages 141-151.

use displayed on pages 410-417 on the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be found
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• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms”: “Exercise Problems 1, 2, 6, and 8” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Isomorphisms”: “Exercise Problems 1, 2, 6, and 8” (PDF)

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Instructions: Work on problems 1, 2, 6, and 8 on page 151.  Then, check your answers against the solutions on page 401.

2.6 Cosets, Normal Subgroups, and Factor Groups   2.6.1 Cosets   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and LaGrange’s Theorem” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and LaGrange’s Theorem” (PDF)

Also available in:

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Instructions: For the section on examples of sets, read “6.1
Cosets,” pages 92-94.

use displayed on pages 410-417 on the PDF file. The material linked
version can be found
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• Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E-222 Abstract Algebra”: “Equivalence Relations; Cosets; Examples” Link: Harvard University Extension: Dr. Benedict Gross’s “Math E-222 Abstract Algebra”: “Equivalence Relations; Cosets; Examples” (YouTube)

Also Available in: Adobe Flash, Quicktime, or Mp3

Instructions: Please click on the link, and watch the video in its entirety.  To choose another format, scroll down to Week 2 and choose the format most appropriate for your internet connection to download the second lecture listed in Week 2 titled “Equivalence Relations; Cosets; Examples.”  Please watch the entire video (about 47 minutes) to learn about equivalence relations and how they are related to cosets.

Terms of Use: This video has been reposted with permission for non-profit educational use by Harvard University.  The original version can be found here.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and Lagrange's Theorem”: “Exercise Problems 1 and 5” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Cosets and Lagrange's Theorem”: “Exercise Problems 1 and 5” (PDF)

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Instructions: Try to do problems 1 and 5 on page 98.  Then, check the solutions on page 399.

2.6.2 Normal Subgroups and Factor Groups   - Reading: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups”  (PDF)

Also available in:

``````[EPUB](https://resources.saylor.org/wwwresources/archived/site/wp-content/uploads/2011/08/MA231-1.1.1book-Thomas-W.-Judson.epub)

Instructions: For the section on examples of sets, read “Chapter
10: Normal Subgroups and Factor Groups,” pages 155-161.

use displayed on pages 410-417 of the PDF file.  The material linked
attributed to Thomas W. Judson and the original version can be
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