What is Abstract algebra: Definition and 457 Discussions

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures 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.
Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.

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  1. Infrared

    Challenge Math Challenge Thread (October 2023)

    The Math challenge threads have returned! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Do not solve problems that are way below your level. Some problems...
  2. Codezero

    I Modeling Asteroid Rotation Using Quaternions: Seeking Guidance on Init

    Hello everyone, I am an International Baccalaureate (IB) student working on my extended essay, which is a mandated 4,000-word research paper. My chosen topic is Quaternions, a mathematical concept I find highly intriguing. The primary aim of my paper is to model the rotation of an asteroid...
  3. I

    What is meant by compex dimension? (Abstract algebra)

    picture since the text is a little hard to read i have no problem showing this is a vector space, but what is meant by complex dimention? Is it just the number on independant complex numbers, so n?
  4. M

    I Algebra Homomorphisms as Subsets of the Cartesian Product

    Let ## \varphi \subseteq A \times B; \psi \subseteq B \times C ##. Then ## \varphi \circ \psi = \left \{ (a, c)| \exists b: (a,b) \in \varphi, (b,c) \in \psi \right \} \subseteq A \times C##. Task: Let ##\varphi## and ##\psi## are subalgebras of algebras ##A \times B## and ##B \times C##...
  5. chwala

    I Understanding the operation in ##(\mathbb{z_6})^{*}##

    ...Out of interest am trying to go through the attached notes, My interest is on the highlighted, i know that in ##\mathbb{z}/\mathbb{6z}## under multiplication we shall have: ##1*1=1## ##5*5=1## am assuming that how they have the ##(\mathbb{z}/\mathbb{6z})^{*}={1,5}## is that correct...
  6. S

    B Commutative diagram conventions

    I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way? In the...
  7. A

    I What is the size of the quotient group L/pZ^m?

    Hello, I have a question that I would like to ask here. Let ##L = \left\{ x \in \mathbb{Z}^m : Ax = 0 \text{ mod } p \right\}##, where ##A \in \mathbb{Z}_p^{n \times m}##, ##rank(A) = n##, ## m \geq n## and ##Ax = 0## has ##p^{m-n}## solutions, why is then ##|L/p\mathbb{Z}^m| = p^{m-n}##? I...
  8. H

    I Is the Order of an Automorphism in a Field with Characteristic p Equal to p?

    Please, I have a question about automorphism: Let ##\mathbb{K}## be a field, if ##\operatorname{char}(\mathbb{K})=p ##, then the order of automorphism ##\phi## is ##p##, i.e. ##\phi^p=\operatorname{id}##, where ##i d## is identity map. Is that right? please, if yes, how we can prove it, and...
  9. PragmaticYak

    Fixed point free automorphism of order 2

    I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
  10. A

    What benefits can this site offer to increase knowledge and understanding?

    Hi all, It is nice to be a member in this site! Hope it will be beneficial and add to my knowledge and understanding.
  11. J

    I Show ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##

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  12. P

    Book recommendations: Abstract Algebra for self-study

    Hello, I am looking for one or more books in combination for self-study of abstract algebra. Desirable would be a good structure of the book with good examples of sentences and definitions. Of course, exercise problems should not be missing. I am now almost tending to buy the Algebra 0 book by...
  13. P

    A Proof of the inequality of a reduced basis

    I would like to show that a LLL-reduced basis satisfies the following property (Reference): My Idea: I also have a first approach for the part ##dist(H,b_i) \leq || b_i ||## of the inequality, which I want to present here based on a picture, which is used to explain my thought: So based...
  14. P

    I Exploring the Relationship Between Lattice Successive Minima and Basis Vectors

    Hello, I've been thinking a bit about the definition of the ##i##-th successive minima of a lattice (denoted with ##\lambda_i(\Lambda)##), and I would argue that the ##i##-th successive minimum is at most as large as the largest lattice basis vector ##b_i##. More formally...
  15. malawi_glenn

    Other Collection of Free Online Math Books and Lecture Notes (part 1)

    School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑 Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...
  16. D

    I Irreducible polynomials and prime elements

    let p∈Z a prime how can I show that p is a prime element of Z[√3] if and only if the polynomial x^2−3 is irreducible in Fp[x]? ideas or everything is well accepted :)
  17. N

    MHB Abstract algebra: i need examples of ...

    please offer me examples of: a) 3 vector spaces over the same field; and b) the same vector space over 3 fields.
  18. K

    What is the geometric approach to mathematical research?

    I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...
  19. V

    A Equivalence Relation to define the tensor product of Hilbert spaces

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  20. EchoRush

    I This would be a false statement, correct?

    I believe that I am correct, the following statement here must be FALSE, right? It has to be false because A union B is like the two entire circles of the Venn diagram and that cannot be a subset of the intersection area, right? Now if this statement was flipped, then it would be true?
  21. C

    MHB Either or statement from Abstract Algebra Book

    Dear Everyone, What are the strategies from proving a either-or statements? Is there a way for me to write an either-or statement into a standard if-then statements? For example, this exercise is from Dummit and Foote Abstract Algebra 2nd, "Let $x$ be a nilpotent element of the commutative...
  22. J

    Courses Which Abstract Algebra Sequence is Best for Aspiring High Energy Physicists?

    My university offers two different two-semester sequences for learning abstract algebra, and I can't decide which one would be better for me, a physics major. Here are the two sequences and their course descriptions, copied and pasted from the university website: Algebra 1: Theory of groups...
  23. Prof. 27

    Finding Cosets of subgroup <(3,2,1)> of G = S3

    Homework Statement Find all cosets of the subgroup H in the group G given below. What is the index (G : H)? H = <(3,2,1)>, G = S3 Homework EquationsThe Attempt at a Solution I will leave out the initial (1,2,3) part of the permutation. We have S3 =...
  24. T

    I Compatibility of Physics and Abstract Algebra

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  25. S

    How can I prove that these relations are bijective maps?

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  26. W

    I Polynomial Ideals: Struggling with Ring Ideals

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  27. W

    I Understanding Quotient Groups in Abstract Algebra

    So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following: Consider the Klein four group, if we...
  28. Math Amateur

    MHB Solves Theorem 3.2.19 in Bland's Abstract Algebra

    I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with another aspect of the proof of Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows...
  29. S

    A Number Line in Synthetic differential geometry

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  30. N

    I Rings, Modules and the Lie Bracket

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  31. S

    A A decreasing sequence of images of an endomorphisme

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  32. A

    I Proving that an action is transitive in the orbits

    <Moderator's note: Moved from General Math to Differential Geometry.> Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point. Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...
  33. The Bill

    I What are the groups for NxNxN puzzle cubes called?

    The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group. What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term? I've been trying to find literature on the groups for the...
  34. U

    Determining a group, by checking the group axioms

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  35. L

    I How many generators can a cyclic group have by definition?

    Hi, so I have just a small question about cyclic groups. Say I am trying to show that a group is cyclic. If I find that there is more than one element in that group that generates the whole group, is that fine? Essentially what I am asking is that can a cyclic group have more than one generator...
  36. mr.tea

    I Help Needed: Understanding Hungerford's Algebra Book Proofs

    I am trying to learn about free groups(as part of my Bachelor's thesis), and was assigned with Hungerford's Algebra book. Unfortunately, the book uses some aspects from category theory(which I have not learned). If someone has an access to the book and can help me, I would be grateful. First...
  37. A

    I How can there only be two possible four-element groups?

    How can you prove that there can only be 2 possible four-element group?
  38. R

    Courses Is proof based Linear Algebra be similar to Abstract Algebra

    I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract...
  39. B

    B What do "linear" and "abstract" stand for?

    What does "linear" in linear algebra and "abstract" in abstract algebra stands for ? Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##...
  40. M

    Show isomorphism under specific conditions

    Homework Statement Let ##A,B## be subgroups of a finite abelian group ##G## Show that ##\langle g_1A \rangle \times \langle g_2A \rangle \cong \langle g_1,g_2 \rangle## where ##g_1,g_2 \in B## and ##A \cap B = \{e_G\}## where ##g_1 A, g_2 A \in G/A## (which makes sense since ##G## is abelian...
  41. mr.tea

    Algebra Question Regarding Purchasing an Algebra Book

    Hi, I am a math undergraduate major and just finished my first abstract algebra course. Unfortunately, we used the lecturer's notes which are quite dry, without motivation, and it really felt bad. I am really interested in abstract algebra, and thus has decided to re-learn it over the summer...
  42. Z

    Graphics using Abstract Algebra

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  43. Mr Davis 97

    Abstract algebra class equation

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  44. BubblesAreUs

    Algebra Textbook for Abstract Algebra / Group Theory

    I am looking for an accessible textbook in group theory. The idea here is to use it to learn basic group theory in order to take up Galois Theory. My background includes Calculus I-IV, P/Differential Equations, Discrete Mathematics including some graph theory, Linear algebra, and am currently...
  45. Yiming Xu

    I Express power sums in terms of elementary symmetric function

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  46. G

    Group is a union of proper subgroups iff. it is non-cyclic

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  47. S

    I First Sylow Theorem: Group of Order ##p^k## & Cyclic Groups

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  48. N

    A very very hard college algebra problem

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  49. P

    B Sets and functions that gain more structure with context

    So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it...
  50. B

    Algebra What are the best books for understanding universal algebra?

    What are good books in universal algebra, given that I have a background in Herstein (Topics in Algebra), Hubbard/Hubbard, Engelking (Topology), and Dugundji (Topology)? I am currently reading Hungerford, and I found a field called universal algebra while searching internet for some concepts...
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