Abstract Algebra

Definition

Abstract [6] algebra is a branch of pure mathematics that studies algebraic structures such as groups, rings, and fields. Far from the rote manipulations you might recall from high school, this area delves into deep, foundational aspects of mathematics and underpins a myriad of applications, from cryptography and computer science to physics and engineering. [1]

Abstract algebra is the branch of algebra dealing with the study of algebraic systems or structures with one or more mathematical operations associated with elements with an identifiable pattern, differing from the usual number systems. In abstract algebra, the elements combined to perform mathematical operations are not interpretable as numbers, hence its abstract nature.

The elements of abstract algebra operate as an abstraction of the algebraic properties common to different number systems and other objects of mathematical study. Therefore, its objective is to learn about the properties of operations, regardless of the operands’ characteristics. Most of this branch was created in the 19th century to respond to the need for greater accuracy in mathematical definitions. [2]

Who

Abstract algebra also has heavy application in physics and computer science through the analysis of vector spaces. For example, the Fourier transform and differential geometry both have vector spaces as their underlying structures; in fact, the Poincare conjecture is (roughly speaking) a statement about whether the fundamental group of a manifold determines if the manifold is a sphere. [3]

Abstract Algebra deals with groups, rings, fields, and modules. These are abstract structures which appear in many different branches of mathematics, including geometry, number theory, topology, and more. They even appear in scientific topics such as quantum mechanics. [Abstract Algebra]

Unless you’re in an MS in pure mathematics or a PhD, you probably won’t find a “need” to take it. It’s useful as an introductory course to abstraction in mathematics (usually the first course students encounter, which helps them stretch their ability to understand things that don’t explicitly exist in the real world), and much of group theory and category theory originate in abstract algebra, as do parts of algebraic topology and algebraic geometry. As mentioned, real analysis and abstract algebra are foundational to developing skills needed in subsequent classes. I’d argue high school geometry/algebra require a different approach than prior math courses, as does calculus. Real analysis/abstract algebra probably make up the next logical jump in mathematics education for now. I’m sure it will be something else (maybe category theory) will be added in the next century, as math becomes even more abstracted. [5]

What

What can abstract algebra be used for?

The main purpose of abstract algebra is analyzing a set endowed with one or more operations with special characteristics or properties to learn about the relationships between those properties of the operations in a precise way, as well as the consequences and possible results of their associations. [2]

What are algebraic structures?

In abstract algebra, “structure” means a set or group with one or more algebraic operations. These structures are classified according to the number of operations that may exist in them, as well as by their characteristics, the number of elements or subsets, and the relationship that exists among the elements of the main set, regardless of their nature. [2]

What is the Law of Composition?

This is a term from abstract algebra used to name a type of binary operation where two elements of given sets are assigned to another element, giving rise to distinct algebraic structures. The law of composition can be internal or external, depending on whether the elements in the application are part of the same set or of different sets, respectively.

The internal laws of composition are represented by the following symbols: ⊙, ⊚, ⊕, ⊖, ⊛, ⊗, and ⊘. The external laws of composition are represented by the following symbols: ⋅, ∘, +, -, ⋇.x, and ∕. The sets are represented by capital letters (A, B, C…) and their elements with lowercase letters (a, b, c…). [2]

What types of algebraic structures are there?

The most common algebraic structures from the abstract branch can be classified as: [2]

A single law of composition or binary operation:
- Magmas: algebraic structures of the form (A,⊚) where A is a set with a single internal binary operation.
- Semigroups: structures of the form (A,⊚) where A is a non-empty set and ⊚ is an internal operation defined in A.
- Groups: algebraic structures formed by a non-empty set with an internal operation that combines a pair of elements to compose a third element within a set.
- Quasigroups: algebraic structures with a linear term that are configured like a magma with a single law of internal composition whose elements are divisible. Their main difference from groups is that they are not necessarily associative.
- Monoids: algebraic structures with associative operations and a neutral element; the latter is what differentiates them from semigroups.
Has two or more laws of composition or binary operations:
- Rings: algebraic systems composed of a set and two internal binary operations that are expressed as (R,+,∙).
- Bodies (also called fields): they are commutative rings of division.
- Modules: algebraic structures that act in group representation theory, where a group entails concrete transformations of a mathematical object.
- Vector spaces (also called linear spaces): algebraic structures composed of a non-empty set, an internal operation (called a sum), and an external operation (called a scalar product). The elements of a vector space are called vectors.
- Associative algebras: they are modules that also allow the multiplication of vectors in a distributive and associative way.
- Lie algebras: algebraic structures defined on a vector space and normally associated with the Lie groups. This mathematical object was previously called an infinitesimal group.
- Lattices: they are algebraic structures used in algebra and order theory; their name comes from the shape of Hasse diagrams.
- Boolean algebras: algebraic structures that outline logical operations, used both in mathematics and in digital electronics and computer science.

What is the difference between abstract algebra and elementary algebra?

While elementary and abstract algebra both respond to the same general approaches to algebra, there are differences between them; for example, while elementary algebra is based on solving simple algebraic equations, abstract algebra looks at algebraic systems and structures or groups with different operations. Elementary algebra studies real numbers and complex numbers, while abstract algebra expresses mathematical structures that cannot necessarily be expressed with numerical values.

Why

See Theoretical Knowledge Vs Practical Application.

*A Conceptual Introduction to Abstract Algebra*

How

Many of the References and Additional Reading websites and Videos will assist you with understanding and applying abstract algebra.

As some professors say: “It is intuitively obvious to even the most casual observer.“

References

[1] Johnson, Andrew. 2023. “Unveiling the Mysteries of Abstract Algebra: A Gateway to Modern Mathematics.” Medium. August 31. https://medium.com/@andrew_johnson_4/unveiling-the-mysteries-of-abstract-algebra-a-gateway-to-modern-mathematics-e8e1d1b2a110.

When people hear the term “algebra,” they often think of solving equations for unknowns — like finding x in 2x+3=7. However, there is an entire universe of algebra that transcends these rudimentary calculations: the field of abstract algebra.

[2] “Abstract Algebra: What Is It, Uses, Structures, Types.” 2022. ferrovial. November 17. https://www.ferrovial.com/en/stem/abstract-algebra/.

[3] “Abstract Algebra.” 2023. Brilliant Math & Science Wiki. Accessed October 12. https://brilliant.org/wiki/abstract-algebra/.

Roughly speaking, abstract algebra is the study of what happens when certain properties of number systems are abstracted out; for instance, altering the definitions of the basic arithmetic operations result in a structure known as a ring, so long as the operations are consistent.

For example, the 12-hour clock is an example of such an object, where the arithmetic operations are redefined to use modular arithmetic (with modulus 12). An even further level of abstraction–where only one operation is considered–allows the clock to be understood as a group. In either case, the abstraction is useful because many properties can be understood without needing to consider the specific structure at hand, which is especially important when considering the relationship(s) between structures; the concept of a group isomorphism is an example.

[4] Amit, Alon. “What are some real-life applications of abstract algebra?”. 2023. Accessed October 12. https://qr.ae/pKh28Q.

[5] “What are some real-life applications of abstract algebra?”. 2023. Quora. Accessed October 12. https://www.quora.com/What-are-some-real-life-applications-of-abstract-algebra.

[6] McNulty, Keith. 2023. “What Does ‘Generalization’ and ‘Abstraction’ Mean in Math?” Medium. Medium. October 12. https://keith-mcnulty.medium.com/what-does-generalization-and-abstraction-mean-in-math-5bc3dc6d513a.

Fully developed professional mathematicians, however, are trained to generalize and abstract their thinking wherever possible. Instead of working with integers or established everyday number structures, they will work with abstracted algebraic structures such as groups, rings, modules or fields, which encapsulate many known, more specific structures, allowing their results to be more powerful because they apply to more general structures and problems.

Additional Reading

⭐ “Abstract Algebra.” 2023. Socratica. Accessed October 12. https://www.socratica.com/courses/abstract-algebra.

“Abstract Algebra.” 2023. Wikipedia. Wikimedia Foundation. August 5. https://en.wikipedia.org/wiki/Abstract_algebra.

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.

“Abstract and Geometric Algebra.” 2022. Mathematics LibreTexts. October 19. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra.

Abstract algebra is the study of algebraic structures and include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.

“Abstract vs. Linear Algebra: Unraveling the Difference.” 2023. Free Science Information. July 8. https://freescience.info/abstract-vs-linear-algebra-unraveling-the-difference/.

Are you ready to dive into the fascinating world of abstract and linear algebra? These elementary branches of mathematics unlock the secrets behind complex mathematical structures, providing a framework that helps us make sense of the most intricate concepts. Abstract algebras delve into the study of various algebraic structures, including commutative algebras, while linear algebra focuses on vector spaces and linear transformations. From matrices and algebraic equations to linear equations and quadratic equations, these fields offer powerful theorems for understanding number systems and solving real-world problems.

Cole. 2023. “What Is Abstract Algebra?” Medium. Cantor’s Paradise. July 26. https://www.cantorsparadise.com/what-is-abstract-algebra-2658c873498e.

Abstract algebra is one of the foundational fields of modern mathematics. It contains a wide variety of subfields and has an immense number of applications. More specifically, abstract algebra is the study of algebraic structures, which are all sorts of things like groups, rings, fields, modules, magmas, and many more! In this article, I am going to discuss the history of abstract algebra, introduce you to some of its basic concepts, and provide some links to learn more about this fascinating area of mathematics.

McNulty, Keith. 2023. “A Conceptual Introduction to Abstract Algebra.” Medium. Medium. December 21. https://keith-mcnulty.medium.com/a-conceptual-introduction-to-abstract-algebra-1b7bff616f2a.

Most people who have not studied Mathematics at higher levels understand algebra to be anything involving equations that have unknowns in them. But as the study of Math becomes more advanced, algebra as a concept becomes more well defined. In essence, algebra is the study of sets of discrete objects that adhere to certain defined operations or rules. The sets and rules are usually constructed to model useful things we see in the real world. For example, if we wish to model binary logic, we could study the set {0,1} and apply AND and OR rules to it. If we wish to model certain transformations in 2D space, we could study the set of 2×2 matrices under matrix addition or multiplication.

Thomas W. Judson, Stephen F. Austin State University via Abstract Algebra: Theory and Applications. 2022. “Abstract Algebra: Theory and Applications (Judson).” Mathematics LibreTexts. March 6. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%3A_Theory_and_Applications_(Judson).

Theory and Applications is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.

Ernst, Dana. 2022. “1.1: What Is Abstract Algebra?” Mathematics LibreTexts. Libretexts. April 17. https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/01%3A_Introduction/1.01%3A_What_is_Abstract_Algebra.

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. This course is an introduction to abstract algebra. We will spend most of our time studying groups. Group theory is the study of symmetry, and is one of the most beautiful areas in all of mathematics. It arises in puzzles, visual arts, music, nature, the physical and life sciences, computer science, cryptography, and of course, throughout mathematics. This book covers the basic concepts of group theory, and a special effort will be made to emphasize the intuition behind the concepts and motivate the subject matter. In the final chapter, we will also introduce rings and fields.

Videos

What is Abstract Algebra? (Modern Algebra)

Abstract Algebra is very different than the algebra most people study in high school. This math subject focuses on abstract structures with names like groups, rings, fields and modules. These structures have applications in many areas of mathematics, and are being used more and more in the sciences, too.

“Abstract Algebra.” 2023. YouTube. YouTube. Accessed October 12. https://www.youtube.com/playlist?list=PLO1y6V1SXjjNZ90TB7r2lGW-ZFPlCZ1lO. (Play List from The Math Sorcerer)

Teaching myself abstract algebra

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

Medium Member Only

The featured image on this page is from Oates, Mark. 2023. “Free Floating Cubes On Blue Image: Stunning Photography.” Burst. Accessed October 12. https://www.shopify.com/stock-photos/photos/floating-cubes-on-blue.