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> =
{g^{n} 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 11 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.*
Unit 2 Time Advisory
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:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read “3.2
Definitions and Examples,” pages 4047.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
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.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.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)
Also available in:Instructions: Work through problems 2, 8, 15, and 17 on page 51. After you complete each exercise, please see the solutions on page

Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).

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.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
Reading: Wikipedia: Finite Groups Link: Wikipedia: “Finite Group” (PDF)
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.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).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 nowfamous piece of group theory humor set to a fourpart 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).
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.
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:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read Chapter 4,
“Cyclic Groups”, pages 5773.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410 417 of the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
Reading: Knowledgerush: “Cyclic Group” Link: Knowlegerush: “Cyclic Group” (PDF)
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.
Terms of Use: The material linked above is licensed under the GNU Free Documentation License (HTML). The original version can be found here (HTML).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)
Also available in:Instructions: Complete problems 3 and 4 on page 69. After you have finished each problem, check the solutions on page 397.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributid to Thomas W. Judson and the original version can be found here (PDF).
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:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read Chapter 5
“Permutation Groups”, pages 7491.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
Lecture: YouTube: “Symmetries of a star and Its Permutation Group” Link: YouTube: “Symmetries of Star and Its Permutation Group” (YouTube)
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.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
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.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
 Reading: Wikipedia: “Symmetric Group”
Link: Wikipedia: “Symmetric
Group”
(PDF)
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.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
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.
Terms of Use: Please respect Wolfram MathWorld's terms of
use. MathWorld
webpages are free for academic use and may be hyperlinked, according
to their FAQ
site.
 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.
Terms of Use: The article above is released under a Creative Commons AttributionShareAlike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).
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/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read “3.3
Subgroups,” pages 4750.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 on the PDF file. The material linked
above is licensed under the [GNU Free <span
style="display: none; "> </span>Documentation<span
style="display: none; "> </span>
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
Lecture: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 8” Discussion Link: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 8 ”Discussion (YouTube)
Instructions: Please click on the link, and view the video in its entirety (5:42 minutes) to learn about subgroups.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.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.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
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:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read “Chapter
11: Homomorphisms,” pages 165171.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
Lecture: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 5” Discussion Link: YouTube: VeritySeeker’s “Basic Abstract Algebra, Part 5” Discussion (YouTube)
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 straightforward manner.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.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)
Also available in:Instructions: Try to do problems 2, 4, and 9 on page 173. After you complete the assigned problems, check the solutions on page

Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).

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:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read “Chapter 9:
Isomorphisms,” pages 141151.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
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)
Also available in:Instructions: Work on problems 1, 2, 6, and 8 on page 151. Then, check your answers against the solutions on page 401.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
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:
[EPUB](https://resources.saylor.org/wwwresources/archived/site/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read “6.1
Cosets,” pages 9294.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 on the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). The original
version can be found
[here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
Lecture: Harvard University Extension: Dr. Benedict Gross’s “Math E222 Abstract Algebra”: “Equivalence Relations; Cosets; Examples” Link: Harvard University Extension: Dr. Benedict Gross’s “Math E222 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 nonprofit 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)
Also available in:Instructions: Try to do problems 1 and 5 on page 98. Then, check the solutions on page 399.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 of the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).
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/wpcontent/uploads/2011/08/MA2311.1.1bookThomasW.Judson.epub)
Instructions: For the section on examples of sets, read “Chapter
10: Normal Subgroups and Factor Groups,” pages 155161.
Terms of Use: Please respect the copyright, license, and terms of
use displayed on pages 410417 of the PDF file. The material linked
above is licensed under the [GNU Free Documentation
License](http://www.gnu.org/licenses/fdl.html) (HTML). It is
attributed to Thomas W. Judson and the original version can be
found [here](http://abstract.ups.edu/download/aata20100827.pdf) (PDF).
Lecture: YouTube: “Normal Subgroup Example 1” Link: YouTube: “Normal Subgroup Example 1” (YouTube)
Instructions: Please click on the link, and view the short video, which is approximately 2 minutes, for an example of a normal subgroup.
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: “Normal Subgroups and Factor Groups”: “Exercise Problems 1, 8, and 13” Link: Stephen F. Austin State University: Thomas W. Judson’s Abstract Algebra Theory and Applications: “Normal Subgroups and Factor Groups”: “Exercise Problems 1, 8, and 13” (PDF)
Also available in:Instructions: Work through problems 1, 8, and 13 on page 163. Then, check the solutions on page 402.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410417 on the PDF file. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (PDF).