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.

View More On Wikipedia.org
Recent contents Top users
  1. S

    Proof: a^3 divides b^2 implies a divides b in Abstract Algebra.

    Let a, b be integers a,b>0 show that if a^3 | b^2 then a|b (Consider the prime factorization of a and b) I've tried setting up generic prime factorization of a and b but then don't get any where, I'm not very strong at this subject. Any kind of hints / where to start would help a lot...
  2. C

    How many groups are there of order 7?

    How many groups are there of order 7. I just need to know this to continue with an assignment. Thanks.
  3. O

    Abstract Algebra: Polynomials problem

    Homework Statement Let f(x)=x5-x2-1 \in C and x1,...,x5 are the roots of f over C. Find the value of the symmetric function: (2x1-x14).(2x2-x24)...(2x5-x54) Homework Equations I think, that I have to use the Viete's formulas and Newton's Binomial Theorem. The Attempt at a...
  4. P

    Any good book to review abstract algebra?

    Hi, I am a junior and a math major, and I am almost done with my year-long abstract algebra sequence for undergraduates. While I found the materials interesting, I feel like I got lost at some places in this course, and I would like to review (or in some topics, relearn) the materials that I...
  5. T

    Proving the Center of a Group Generated by x and y is {e, x^n}

    Homework Statement Let G=<x, y| x^{2n}=e, x^n=y^2, xy=yx^{-1}>. Show Z(G)={e, x^n}. Homework Equations The Attempt at a Solution So I tried breaking this up into cases: Case 1: If n=1. then |x|=1 or 2. If |x|=1, then x=e and x would obviously be in the center. If |x|=2, then xy=yx (since...
  6. T

    Abstract Algebra: Find Generators & Relations for Z2+Z2+Z2

    Homework Statement What is the minimum number of generators needed for Z2+Z2+Z2? Find a set of generators and relations for this group. Homework Equations The Attempt at a Solution I think it is obvious that the minimum amount of generators that you need is three, with Z2+Z2+Z2 =...
  7. M

    Understanding in abstract algebra

    In algebra, do you just base your understanding off the pure definitions and groups? I am learning some multilinear algebra, seeing a lot of talk about rings, algebras, modules, etc. and I can't help but thinking it's all just frivolous, pointless definitions. That's partly because I just can...
  8. K

    Find d when d|n2+n-2, d|n3+2n-1 & d=1 (mod 2), d > 1

    Let d=GCD(n2+n-2,n3+2n-1). Find d if d=1(mod 2) & d > 1. So we know d|n2+n-2 & d|n3+2n-1. My question is simply this, the professor wrote down hence d|n3+n2-2n, right after what is written above. But I'm just not seeing how you get that combination. I understand how to work the...
  9. T

    Abstract Algebra Homomorphism Proof

    Homework Statement Let G and H be two groups. If f: G \rightarrow H is a homomorphism, a \in G and b = f(a). If ord(a) = n, ord(b) = m, then n is a multiple of m. (Let e_{1} be the identity of G and e_{2} be the identity of H) I have to prove that n is a multiple of m. Homework Equations...
  10. T

    Abstract Algebra question

    Homework Statement (1)To prove this I have to let G be a group, with |G|=p^2. (2)Use the G/Z(G) theorem to show G must be Abelian. (3) Use the Fundamental Theorem of Finite Abelian Groups to find all the possible isomorphism types for G. Homework Equations Z(G) = the center of G (a is...
  11. T

    Abstract Algebra: Finding Conjugates

    Homework Statement Consider this group of six matrices: Let G = {I, A, B, C, D, K}, Matrix Multiplication> I =\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} A =\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} B =\begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix} C =\begin{bmatrix}-1 & -1\\0 & 1\end{bmatrix} D...
  12. H

    Should I take complex analysis or abstract algebra?

    Being a high school student who will be going into physics, should I take complex analysis or abstract algebra in the fall? I can't take both at once, and I am set to take intro to QM (I will already have taken Calc I-III, an introductory functional analysis course, and linear algebra. I also...
  13. K

    Abstract Algebra, Euclidean Algorithm

    Use the Euclidean Algorithm to find the gcd of the given polynomials: (x3-ix2+4x-4i)/(x2+1) in C[x] First I got x-i R: 3x-3i, then I took the 3x-3i into x2+1 & got 1/3 x R: 1+i. Then I was going to take 1+i into 3x-3i. However that never ends it seems, unless I just confused myself...
  14. C

    Is R={0, 2, 4, 6, 8} a Field under Addition and Multiplication Modulo 10?

    Homework Statement Let R={0, 2, 4, 6, 8} under addition and multiplication modulo 10. Prove that R is a field. Homework Equations A field is a commutative ring with unity in which every nonzero element is a unit. The Attempt at a Solution I know that the unity of R is 6, and that...
  15. K

    Binary Operation, Abstract Algebra

    Define a binary operation on Z, the set of integers by the equation m * n = m + n + mn. Which of the following statemnts is / are true about the binary structure of Z with * 1) * is not associative 2) There is no element e belonging to Z such that for every z belonging to Z, z*e = e*z = z 3)...
  16. E

    Abstract Algebra: Show that 2Z + 5Z = Z

    Homework Statement Show that 2\mathbb{Z} + 5\mathbb{Z} = \mathbb{Z}Homework Equations where 2Z + 5Z = {a+b | a in 2Z and b in 5Z} = ZThe Attempt at a Solution For any n in Z, we can write n= (5-4)n = 5n +(-4)n = 5n + 2(-2n) And since 5n is in 5Z and 2(-2n) is in 2Z, we can form Z from any...
  17. T

    Abstract Algebra: Proving/Disproving |a|=|b| if |a^2|=|b^2|

    Homework Statement If |a^2|=|b^2|, prove or disprove that |a|=|b|. Homework Equations The hint I was given is that let a be an element of order 4n+2 and let the order of b=a2 The Attempt at a Solution I can disprove this by looking at examples, such as in the group Z20 with...
  18. T

    Abstract algebra cyclic subgroups

    Homework Statement Suppose that G is a group with exactly eight elements of order 10. How many cyclic subgroups of order 10 does G have? Homework Equations The Attempt at a Solution I really don't have a clue how to solve this, any help would be greatly appreciated.
  19. icystrike

    Abstract Algebra: Schwarz Inequality Homework

    Homework Statement This is part of the proof of Schwarz inequity. Please help me understand the following equation , i think it should not be a equal sign instead it should be greater or equal to. Homework Equations The Attempt at a Solution
  20. I

    Abstract Algebra: a problem about ideal

    Homework Statement Let J be the set of all polynomials with zero constant term in Z[x]. (Z=integers) a.) Show that J is the principal ideal (x) in Z[x]. b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each n\inZ. Homework Equations The Attempt at...
  21. P

    Help with Plane Lattices Problems in Abstract Algebra

    I have been struggling through this Abstract Algebra class and have completely bogged down in the Wallpaper Patterns chapter, especially the plane lattices section. Can anyone give me some help for the following three problems? I am not sure how to start any of the three problems. Thanks for any...
  22. T

    Is C(a) isomorphic to C(gag-1) for elements a and g in a group?

    Homework Statement If a and g are elements of a group, prove that C(a) is isomorphic to C(gag-1) Homework Equations I have defined to mapping to be f:C(gag-1) to C(a) with f(h)=g-1hg. I have no idea if this is right. The Attempt at a Solution I don't have a clue at the solution...
  23. M

    Abstract Algebra: Solving Stumping Questions | αη = β and G is Abelian

    I have 2 algebra questions which are stumping me, I just can't seem to use my notes to figure them out! 1. Let α, β ∈ S17 where α = (17 2)(1 2 15 17 ), β = (2 3 16)(6 16 17 ). Determine η, as a product of disjoint cycles, where αη = β. 2. Let G be a group in which a^2 = 1 for all a ∈ G...
  24. T

    Abstract Algebra: Proving whether H is a subgroup.

    Homework Statement Let R = {all real numbers}. Then <R,+> is a group. (+ is regular addition) Let H = {a|a \epsilon R and a2 is rational}. Is H closed with respect to the operation? Is H closed with respect to the inverse? Is H a subgroup of G? Homework Equations N/A The Attempt at a...
  25. C

    Abstract algebra. proving things about U(n)

    Homework Statement prove that if g is in Z*_n then g^2=1, so g has order 2 or is the identity. show that the largest value of n for which every non identity element of Z*_n has order 2. which are these others. Homework Equations Z*_n = U(n) different notation it is the the group of co...
  26. J

    Help with abstract algebra proof

    prove that a finite ring with identity has characteristic n for some n>0. been trying for a while getting nowhere any ideas?
  27. T

    Abstract Algebra - Normal groups

    Homework Statement I'll be delighted to receive some guidance in the following questions: 1. Let G1,G2 be simple groups. Prove that every normal non-trivial subgroup of G= G1 x G2 is isomorphic to G1 or to G2... 2. Prove that every group of order p^2 * q where p,q are primes is...
  28. L

    Abstract algebra: Rings and Ideals

    Homework Statement The problem is to show that a subset A of a ring S is an ideal where A has certain properties. S is a ring described as a cartisian product of two other rings (i.e., S=(RxZ,+,*)). I have already proved that A is a subring of S and proved one direction of the definition of an...
  29. O

    Abstract Algebra: Is S(3) Isomorphic to Z(6)?

    Homework Statement Is the symmetric group s(3) isomorphic to Z(6), the group of integers modulo six with addition (mod 6) as its binary operationHomework Equations Basically i know that the symmetric group is all the different permutations of this set and that there are six of them. I also...
  30. D

    Exploring the Intuition Behind Rings in Abstract Algebra

    Ok so I am not a math major and i haven't taken an abstract algebra class but i am curoius about the subject. I have been watching video lectures at UCCS at http://cmes.uccs.edu/Fall2007/Math414/archive.php?type=valid and the proffessor talks about groups and rings. In the introduction the...
  31. W

    Abstract Algebra Concept-based Question

    I have no abstract algebra background (only matrices and calculus and stats) but this problem came up in one of my classes and this time I'm completely clueless: Homework Statement A group is cyclic if an element, g, of the group generates the entire group in the sense that if h is any...
  32. U

    Discussion group for abstract algebra? I'd be interested

    Hi everyone, I've just finished year 11 here in Australia and I've been reading some notes on abstract algebra just out of curiosity. I have had a little difficulty grasping the concepts, and I've read up on some linear algebra (up to the point of Euclidean n-space - haven't yet read about...
  33. P

    Abstract Algebra: Commutative Subgroup

    Homework Statement Let G be a group and let a, b be two fixed elements which commute with each other (ab = ba). Let H = {x in G | axb = bxa}. Prove that H is a subgroup of G. Homework Equations None The Attempt at a Solution I'm using the subgroup test. I know how to show...
  34. P

    I want to get A's in analysis and abstract algebra

    I just finished my first quarter of analysis (Text: Rudin's PMA) and abstract algebra (Text: Beachy and Blair) courses. I must say I really enjoyed these courses, and I feel like I learned a lot from them. However, I still ended up getting B+'s from both of these courses. While I'm not...
  35. K

    Abstract algebra 2 questions

    1. If G is a finite group that does not contain a subgroup isomorphic to Z_p X Z_p for any prime p. prove that G is cyclic im stumped. i don't understand the 'does not contain a subgroup isomorphoc to Z_p X Z_p part. ive tried using cauchy's theorem for abelian group: if G is a finite...
  36. V

    Abstract algebra: systems of differential linear equations

    Homework Statement Solve the inhomogeneous differential equation dX/dt=AX+B in terms of the solutions to the homogeneous equation dX/dt=AX. Homework Equations A is an nxn real or complex matrix and X(t) is an n-dimensional vector-valued function. If v is an eigenvector for A with...
  37. D

    Abstract Algebra: Proving E=F(a^p)

    Homework Statement Let F be a field of characteristic p>0 and let E = F(a) where a is separable over F. Prove that E=F(a^p). Homework Equations The Attempt at a Solution I know that maybe show how mod F(a) = mod F(a^p) or something around there.
  38. V

    Proving Coset Properties in Abstract Algebra

    Question: Prove the following properties of cosets. Given: Let H be a subgroup and let a and b be elements of G. H\leq\ G Statement: aH=bH \ if\ and\ only\ if\ a^{-1}b\ \epsilon\ H The statement is what I have to prove. My issue is I don't know how to start off the problem. When I...
  39. V

    Abstract Algebra - Orbit of a permutation

    For this problem, I have to find all orbits of given permutation. \sigma: \mathbb{Z} \rightarrow \mathbb{Z} Where, \sigma(n)=n-3 Now, the problem is I do not know how to approach this permutation in the given format. All the permutations I dealt with were in the form: \mu...
  40. D

    Abstract Algebra Problem (should be easy)?

    Homework Statement List all the elements of GL_N(\mathbb{Z}/2\mathbb{Z}). Find the order of each element, and show it is not abelian. The Attempt at a Solution I am confused right from the get go about GL_n(\mathbb{Z}/2\mathbb{Z}). I think the L_n(\mathbb{Z}/2\mathbb{Z}) part...
  41. D

    Abstract Algebra: Homomorphism

    Hey all, I`ve been working at this "proof" for several hours now, have put it away several times thinking that maybe I`ll get it if I leave it alone for a bit...has not worked =] It has 2 parts, I think I have proven the first part, but the second one really just stumps me =| 1. Show that a...
  42. B

    Abstract algebra question chapter 1.2

    Homework Statement prove that if 2 does not divide a then 24 divides a^2-1 Homework Equations I know that if 2 does not divide a then a is odd. I proved that the square for all odd integers are of the form 8K+1 I also proved the square of any integer is either in the form...
  43. J

    Should I take abstract algebra?

    How hard is Artin's Algebra book to understand? For a student who has not had any upper level (proof based) math classes beyond calculus, is it doable if you are sufficiently motivated?
  44. M

    Abstract algebra - direct sum and direct product

    Hi everybody, I'm new to absract algebra and I really can not understand different between direct sum and direct product in group theory (specially abelian groups). could does anyone give me a clear example or ... ? thanks
  45. P

    Abstract Algebra & Computer Science

    Hi, I have heard a few times that it is beneficial to study abstract algebra if I want to study computer science at advanced level (i.e. upper class, grad school, etc.), but is this true? If so, why would it be so? Thanks
  46. F

    Difficult choice regarding Abstract Algebra

    Hi all. I am currently a Junior attending SUNY-Stony Brook as a math major. This coming fall semester I have a very good opportunity before me: I can either take the undergrad abstract algebra course (textbook: Contemporary Abstract Algebra by Gallian), or I can take the graduate abstract...
  47. M

    Develop Intuition with Abstract Algebra: Groups & Rings

    I've taken 2 (undergrad) courses in abstract algebra and a reading course in Galois Theory, and I still don't understand the point of studying groups and rings. The courses have not been particularly difficult for me, but my motivation is extremely low. In Galois Theory obviously I saw an...
  48. G

    Abstract Algebra, I don't understand what my HW question is asking

    Abstract Algebra- VERY SIMPLE but I don't understand what my HW question is asking! Homework Statement Hi. I am having trouble simply understanding what the question is here: (6) let w = (1 2 3 4 5 6 7 8 9 10 11 12 13 14). For which integers i is w^i a 14-cycle? Here is a link...
  49. H

    Abstract Algebra: Ring Theory Problems

    Hello all, I am trying to work on some Ring Theory proofs and my Abstract Algebra is very minimal as I have not taken the class but need to look into it nonetheless. If anyone can figure these out for me I'd greatly appreciate it. Also, I am familiar in LaTeX typesetting but I don't know...
  50. B

    Understanding Abstract Algebra: A Geometric Approach

    I'm taking a class in abstract algebra this summer, so I thought I'd get ahead by reading the book before class starts. This is from a book called "Abstract Algebra: A Geometric Approach", chapter 1: Applying the Principle of Mathematical Induction with a slight modification. If S' \subset \{n...
Back
Top