# MA232: Abstract Algebra II

Unit 1: Advanced Discussion on Groups   We will begin this unit with a series of counting methods involving groups.  We will start with *Lagrange’s Theorem.  Joseph Lagrange discovered the interesting fact that the order, or number of elements, of any true subgroup of a finite group will divide the order of the main group.  This is important to know: if a group’s order is prime, then the group is a cyclic, simple group, containing no nontrivial subgroups.*

Remembering that a symmetric group is the collection of all permutations of a set X, we can think of a *
group action
as a group homomorphism from some group G to the symmetric group of X if the mapping from G’s identity element to the permutation group of X is the identity transformation and the mapping from gh in G is the composition of permutations assigned to g and h.  That is, if we think of X as a set of points on some regular geometric figure, then the group action is a way of describing rotations and reflections of the figure.  Group actions are powerful; they allow geometric ideas to be applied to abstract structures.*

The *Class Equation** tells us how many elements are in the center of a finite group G (the center being the set of all elements in G that are commutative under the operation on G) and in the distinct cosets consisting of elements of G not in the center.  Remember that by Lagrange’s Theorem, the order of the center must divide the order of G, and if the order of G is prime, then the center is trivial.  If the center contains only the identity element, or the center contains all of G, then G is commutative.  On the other hand, if G contains non-commutative elements and has a nontrivial center, its order cannot be prime.*

Suppose G is a group that acts on a set X.  That is, for every g in G and x in X, gx = y in X.  The set of all permutations of x caused by left products with g is called the *
group orbit** of x.  (Think of x as some coordinate of an object in space and the transformation group G as the group that moves or transforms x around a path.)  Burnside’s counting theorem, also called the Cauchy-Frobenius lemma, tells us exactly how many orbits exist for G and X.  Burnside did not discover this theorem, but credited Ferdinand Frobenius.*

The *
Sylow theorems** involve another counting principle.  Peter Sylow said that every finite group has only so many subgroups of a fixed order.  His theorem tells us how many subgroups of a given order exist.  Further, he said that any subgroup that has a power of a prime order must be the maximal subgroup of that order—that is, such a subgroup cannot be contained in another subgroup of the same order.*

Abelian groups are commutative groups.  They were first studied by Norwegian mathematician Niels Henrik Abel.  Abel discovered that if a group of an equation was commutative, then its roots were solvable by radicals.  After studying abelian groups, we will look at solvable groups, which are constructed from abelian groups using extensions.  From the use of extensions, a group G is called *
solvable** if its derived series eventually reaches the trivial subgroup of identity.  The derived series is a series of commutator subgroups for which commutators are elements of a group G that have the form g-1h-1gh for g, h in G.  It makes sense that the only way g-1h-1gh could reduce to the identity element of G, e, is if g and h are commutative and are hence in the center of G.  The commutator subgroups are not typically commutative, and the larger that the largest commutator subgroup of G is, the “less abelian” G is.*

Nilpotent groups are “almost” abelian through repeated use of commutators.  What we discover is that if a group G is a direct product of its Sylow subgroups, it is also nilpotent.  If G is nilpotent, then every maximal proper subgroup of G is normal.  Nilpotency is an important concept because constructs that are not abelian or solvable may have solvable subconstructs, and there are applications for nilpotent groups in Galois theory.

This unit will take approximately 37 hours to complete.

☐    Subunit 1.1: 4 hours

☐    Subunit 1.2: 5 hours

☐    Subunit 1.3: 5 hours

☐    Subunit 1.4: 4 hours

☐    Subunit 1.5: 6 hours

☐    Video: 1 hour

☐    Assignment: 0.5 hours

☐    Subunit 1.6: 5 hours

☐    Subunit 1.7: 4 hours

☐    Subunit 1.8: 4 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

- Use Lagrange’s theory to determine whether a group is cyclic or not. - Use the Class Equation to determine the size of the center of a nontrivial group. - Use Burnside’s counting theorem to determine the number of orbits of G on some set X. - Use the Sylow Theorems to determine how many subgroups of a finite order are contained in a given finite group. - Use a finite group G’s derived series to determine if G is solvable.

1.1 Lagrange’s Theorem   - 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:

pages 92 – 97.

Note: This PDF file will be used for the entire course.  It will be
referenced for readings and assignments throughout.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be found
``````
• Lecture: VideoLecture.net: Marconi Barbosa’s “Group Theory in Machine Learning” Link: VideoLecture.net: Marconi Barbosa’s Group Theory in Machine Learning (YouTube)

Instructions: Please click on the link above.  The video contains information on applications of group theory to the study of machine learning and this clip shows how Lagrange's Theorem may be applied.

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

`````` Also available in:

223.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````
• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 1, 2, and 3” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 1, 2, and 3” (PDF)

Also available in:

iBooks

Instructions: Do problems 1, 2, and 3 on page 224.  The solutions can be found on page 403.

1.3 The Class Equation   - Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “The Class Equation” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “The Class Equation” (PDF)

`````` Also available in:

213 – 215.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````
• Reading: Wikipedia's “Order (group theory)” Link: Wikipedia’s “Order (group theory)” (PDF)

Instructions: Please click on the link above to read the material.  The Class Equation is on the page along with other discussions of counting methods.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 6, 8, and 11” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 6, 8, and 11” (PDF)

Also available in:

iBooks

Instructions: Do problems 6, 8, and 11 on page 224.  The solutions can be found on page 403.

1.4 Burnside’s Counting Theorem   - Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Burnside's Counting Theorem” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Burnside's Counting Theorem” (PDF)

`````` Also available in:

Theorem, pages 215 – 222.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````

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

`````` Also available in:

– 235.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````
• Lecture: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Alternating group structure” Link: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Alternating group structure” (Flash, QuickTime or Audio mp3)

Instructions: Please click on the link, then scroll down to Week 8, Lecture 1.  Choose the format most appropriate for your internet connection.  Watch the entire video.  This video clip discusses alternating group structures, but begins with a discussion on the Sylow theorems.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems”: “Exercise Problems 1, 2, and 17” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems”: “Exercise Problems 1, 2, and 17” (PDF)

Also available in:

iBooks

Instructions: Do problems 1, 2, and 17 on page 235 – 236.  The solutions can be found on page 404.

1.6 Abelian Groups   - Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Finite Abelian Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Finite Abelian Groups” (PDF)

`````` Also available in:

201.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````

• Lecture: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Review, Kernels, Normality” Link: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Review, Kernels, Normality” (Flash, QuickTime, or Audio mp3)

Instructions: Please click on the link, then scroll down to Week 2, Lecture 1.  Choose the format most appropriate for your internet connection.  Watch entire video.

Note on the Media: The video clip discusses kernels, normalities and centers.

• Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 1, 4, and 7” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 1, 4, and 7” (PDF)

Also available in:

iBooks

Instructions: Do problems 1, 4, and 7 on page 205 – 206.  The solutions can be found on page 403.

1.7 Solvable Groups   - Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 12, 16, and 21” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 12, 16, and 21” (PDF)

`````` Also available in:

Instructions: Do problems 12, 16, and 21 on pages 206 – 207.  The
solutions can be found on page 403.

use displayed on pages 410 – 417.  The material linked above is
licensed under the [GNU Free Documentation
attributed to Thomas W. Judson and the original version can be
``````
• Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Solvable Groups” Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Solvable Groups” (PDF)

Also available in:

iBooks