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

    What's the Next Step in Advanced Algebra After Artin and Dummit?

    I just finished a very rigorous second course in linear algebra covering determinants, diagonalization, cayley hamilton thm and invariant subspaces, normal/self adjoint/unitary operators and the spectral thm, and jordan forms. I also have finished calc 3, analysis in several dimensions. I...
  2. mnb96

    Co/contra/in-variance of tensors in abstract algebra

    Hello, The concept of contravariance, covariance and invariance are commonly used in the domain of Tensor Calculus. However I have heard that such concepts are more abstractly defined (perhaps) in cathegory theory. Could someone explain shortly the connection between the abstract definitions...
  3. E

    Abstract Algebra modular Arithmatic Proof

    Homework Statement Prove 10n ≡18 10 for all n ϵ N Homework Equations I have no idea where to even begin this proof. The Attempt at a Solution
  4. B

    Abstract Algebra: Prove a non-abelian group of order 10 must have an elemnt of order2

    Homework Statement Prove that a non-abelian group of order 10 must have an element of order 2. What if the order of every element is 5? Prove there are 5 elements of order 2.
  5. G

    Ideal Rings - Abstract Algebra

    Homework Statement Suppose R is a ring and I,J is an ideal to R. Show (i) I+J is ideal to R. (ii) I union J is ideal to R.Homework Equations none
  6. E

    Divisibility Proof (Abstract Algebra)

    Homework Statement let a belong to N and x,r belong to Z use the definition of divisibility along with the axioms of Integers to prove that IF 5|a and 15|(2ax+r) then 5|r Homework Equations How do I continue the proof?? The Attempt at a Solution So I have: let a belong to N and...
  7. K

    Applications of Abstract Algebra?

    What are some applications of abstarct algebra? I have to write a paper and present on a application of abstract algebra and am looking for topic ideas.
  8. T

    Abstract Algebra - Subgroups of index 2 in R*

    Homework Statement I am having trouble understanding how I would go about finding all subgroups of index 2 in R*, the multiplicative group of nonzero real numbers. Any hints will be greatly appreciated. Homework Equations The Attempt at a Solution
  9. K

    Permutation Expressions: Understanding and Computing

    Homework Statement Compute the expression shown for the permutations 1.\left|<\phi>\right| 2..\left|<\tau^2>\right| 3.\phi^{100} where: \phi= top row:1, 2 , 3 ,4 , 5 ,6 bottom row: 3,1, 4,5,6,2 \tau = top row: 1,2,3,4,5,6 bottom...
  10. H

    Abstract algebra / binary operation

    Homework Statement a. In each case a binary operation * is given on a set M. Decide whether it is commutative or associative, whether an identity exists, and find the units. M=N(natrual); m*n = max(m,n) b. If M is a moniod and u in M, let sigma: M -> M be defined by sigma(a) = ua for all a...
  11. K

    Abstract Algebra - roots of unity

    Homework Statement I want to find out if the sixth root of unity is a subgroup of the complex numbers with multiplication. Homework Equations The Attempt at a Solution I know it's true but my problem is getting there. I know the sixth root of unity must be closed under the...
  12. K

    Abstract Algebra Mod 6 Subgroup Computation and Generator Identification

    Homework Statement I'm working with a mod 6 addition table. I want to compute the subgroups <0>,<1>,<2>,<3>,<4>,<5> I also want to find what elements are generators of the group mod 6. Then I wnat to use do a subgroup diagram. Homework Equations The Attempt at a Solution I am...
  13. K

    Abstract Algebra - showing commutativity and associativity

    Homework Statement I need to determine if a*b=ab+1 is commutative and associative. * is any arbitrary operation Homework Equations The Attempt at a Solution a*b=ab+1 is commutative, but not associative. I'm getting stuck in showing why. b*a=ba+1 a*(b*c)=a(b+1) ?
  14. D

    Abstract Algebra Proof (factorization?)

    Homework Statement If n is a nonzero integer, prove that n cam be written uniquely in the form n=(2^k)m, where k is greater than or equal to zero, and m is odd Homework Equations It is in the primes and unique factorization chapter so maybe that every integer n (except 0 and 1) can be...
  15. Z

    How can induction be used to prove that a function is a numerical polynomial?

    Def: A polynomial f(x) with coefficients in Q (the rationals) is called a "numerical polynomial" if for all integers n, f(n) is an integer also. I have to use induction to prove that for k > 0 that the function f(x) := (1/k!)*x*(x-1)...(x-k+1) is a numerical polynomial I checked that...
  16. D

    Introduction to Group Theory - Abstract Algebra

    Homework Statement Prove that if (ab)2 = a2b2 in a group G, then ab = ba.Homework Equations * For each element a in G, there is an element b in G (called the inverse of a) such that ab = ba = e (the identity). * For each element in G, there is a unique element b in G such that ab = ba = e. *...
  17. U

    Is Abstract Algebra a Necessary Foundation for Mathematical Maturity?

    Hi everyone, I'm in Grade 11 this year (in Australia), currently studying from Apostol's first volume "Calculus". I have just recently started working on the theory of integration of trigonometric functions (just giving some information on my background). I am thinking that, perhaps once I've...
  18. B

    Understanding Even Permutations in Abstract Algebra

    [b]1. This is not a question it's an example. [b]2.The permutation (123)= (13)(12)= (13)(23)(12(13)= (23)(13)(12)(13)(12)(23) is even. [b]3. I got the frist one because it is the product of tranposition...I just don't get the rest. I know that it is even depending on the number...
  19. J

    Abstract Algebra problem (related to Rings of Polynomails

    We'll I made it through another semester, but it seems that I am completely stuck on the last problem of the last homework assignment. I've made a little progress, but I'm really having trouble understanding the question. Perhaps someone on these forums will have some insight Homework...
  20. E

    Abstract algebra (ring theory)

    Let R be the set of all a in rational numbers in whose reduced form the denominator is not divisible by a fixed prime p. Verify R is a ring under the usual addition and multiplication in rational numbers. Find all invertible elements in R.
  21. K

    Abstract Algebra: Isomorphic polynomial rings

    Homework Statement If F is an infinate field, prove that the polynomial ring F[x] is isomorphic to the ring T of all polynomial functions from F to F Homework Equations The Attempt at a Solution T is isomorphic to F[x] f(a+b) = f(a) + f(b) f(ab)=f(a)f(b) It is surjective by...
  22. B

    Abstract Algebra Proof question

    Homework Statement The question is: Let A be a subset of Sn that contains all permutations alpha such that alpha can be written as a product of an even number of transpositions. Prove that A is a group with product of permutations. I understand what I need to do to prove it, but I am not...
  23. L

    Abstract Algebra Struggles: Navigating Levi and Malcev Theorems

    Hello! I've got big problems with understanding abstract algebra, the way we deal with it in the seminar on Lie algebras. In just four weeks we progressed up to Levi and Malcev theorems, which are actually the culmination, the say, of classical Lie algebras theory. I didn't think, that the...
  24. H

    Possible Rings with Product of Nonzero Elements Equal to 0?

    for which of the following rings is it possible for the product of two nonzero elements to be 0? 1. ring of complex numbers 2. ring of integers modulo 11 3. the ring of continuous real-valued functions on [0,1] 4. the ring {a+b(sqrt(2)) : a & b are rational numbers} 5...
  25. A

    Question about a Theorem in Gallian's Contemporary Abstract Algebra

    Question about a Theorem in Gallian's "Contemporary Abstract Algebra" I'm using this book as a reference for my Algebra course, and there's a lemma in the book that is really confusing me. It is on Page 102 of the Sixth Edition, for those who have the book. The lemma states: If...
  26. N

    Groups, Normalizer, Abstract Algebra, Dihedral Groups help?

    [b]1. Let G be a Group, and let H be a subgroup of G. Define the normalizer of H in G to be the set NG(H)= the set of g in G such that gHg-1=H. a) Prove Ng(H) is a subgroup of G b) In each of the part (i) to (ii) show that the specified group G and subgroup H of G, CG(H)=H, and NG(H)=G...
  27. S

    Abstract Algebra Test: Can I Still Make an A?

    Hi guys, I was just wondering, i had a test in Abstract Algebra, and i got a 85, which roughly means a B, and i am really pissed off at myself, because if i only had been less stressful during the test i could have easily gotten a score above 90...because not more than 1 hour or sth after the...
  28. T

    Show G(p) of Order p^t when G is an Abelian Group of Order (p^t)m

    If G is an abelian group of order (p^t)m, and (p,m)=1, show that G(p) has order p^t and G(p) = {a e G| |a|=p^m where m is a natural number} any suggestions?
  29. P

    Introductory abstract algebra question (computing permutations)

    Homework Statement Let A, B be permutations and A = (1 3 5 10)(3 15 8)(4 14 11 7 12 9) and B = (1 14)(2 9 15 13 4)(3 10)(5 12 7)(8 11) Find AB. Homework Equations The Attempt at a Solution I am struggling with finding the product of this permutations and can't quite get the...
  30. T

    Exploring Subgroups of the Additive Group Q/Z in Abstract Algebra

    Homework Statement If G is the additive group Q/Z, what are the elements of the subgroup G(2)? Of G(P) for any positive prime P? Where G(n)={a e G| |a| = n^(k) for some k is greater than or equal to 0}...That is the set of all a in G, s.t. the order of a is some power of n. (But since...
  31. T

    Elements of Subgroups in Additive Group Q/Z: G(2) and G(P)

    The question: If G is the additive group Q/Z, what are the elements of the subgroup G(2)? Of G(P) for any positive prime P? Where G(n)={a e G| |a| = n^(k) for some k is greater than or equal to 0}...That is the set of all a in G, s.t. the order of a is some power of n. (But since it is the...
  32. S

    Abstract Algebra Homework: Proving H is a Subgroup of G

    Homework Statement Let a be in a group G, and let H=\{ a^n: n\in Z\}. Show the following: (i) if h and h' are in H, so is hh'. (ii) The identity e of G is in H. (iii) if h is in H, so is h^{-1}. The Attempt at a Solution Here is what i tried. First of all i am not sure...
  33. rocomath

    Can the Rule of Simplification be Applied to Not (P and Q)?

    Abstract Algebra, first Proof :( I really want to do well in this class! :) http://img329.imageshack.us/img329/2636/abstract001sq3.jpg http://img329.imageshack.us/img329/7108/abstract002ym3.jpg Def U = Definition of Universe UQ = Universal Quantification
  34. T

    Abstract Algebra or Real Analysis

    Which one should I take first? Does it help to take one before the other?
  35. T

    What Are the Best Advanced Textbooks for Studying Abstract Algebra?

    My current algebra class is using Fraleigh's "First Course in Abstract Algebra", and it doesn't feel very challenging (and is extremely verbose!). I'd like to study the subject deeper, since I really enjoy it. I picked up Lang's "Undergraduate Algebra", which seems to be much better, but if...
  36. I

    Linear algebra and abstract algebra simultaneously?

    Is this a good idea (provided the university will allow it)? I'll be going into my sophomore year at my university. But I'm unfamiliar with exactly how much linear algebra an intro course in abstract algebra would require. In hindsight I probably should have taken linear last semester, but...
  37. Z

    Abstract algebra: irreducible polynomials

    Homework Statement Prove that f(x)=x^3-7x+11 is irreducible over Q Homework Equations The Attempt at a Solution I've tried using the eisenstein criterion for the polynomial. It doesn't work as it is written so I created a new polynomial...
  38. G

    Abstract Algebra Questions - Need help for exam

    Abstract Algebra Questions - Need help for exam! Homework Statement I am studying Abstract Algebra in college and my exams are approaching fast.I need somebody to help me out to do a few exam papers. I am going to post the questions below from the exam papers and if you can advise me how...
  39. S

    Abstract Algebra (Was: Book recommendation)

    Hi, Next fall i will be taking Intro to Abstract Algebra so i was planning to give it a shot on my own during the summer break, but i don't know what would be a good book to buy online, that is not too expensive. I would like the book to be quite rigorous, like very proof based one, but that...
  40. M

    Proving Isomorphism and Galois Group Existence in Abstract Algebra Homework

    Homework Statement Show that G is isomorphic to the Galois group of an irreducible polynomial of degree d iff is has a subgroup H of index d such that \bigcap_{\sigma \in G} \sigma H \sigma^{-1} = {1} .Homework Equations The Attempt at a Solution I know that if G acts transitively as a...
  41. M

    What are the properties of quartic polynomials with different cubic resolvents?

    Homework Statement I'm trying to come up with an example of a quartic polynomial over a field F which has a root in F, but whose splitting field isn't the same as its resolvent cubic. Homework Equations The Attempt at a Solution Well, I know the splitting field of the cubic...
  42. B

    Groups of Order 144: Abelian Groups Up to Isomorphism

    I'm going insane. The question is: List all abelian groups (up to isomorphism) of order 144. There are 10 non-isomorphic groups of order 144 and I only have 9. Here they are: Z2 X Z2 X Z2 X Z2 X Z3 X Z3 Z2 X Z2 X Z2 X Z2 X Z9 Z4 X Z2 X Z2 X Z3 X Z3 Z4 X Z2 X Z2 X Z9 Z8 X Z2 X Z3 X Z3 Z8 X Z2...
  43. J

    Number Theory & Abstract Algebra

    I'm currently taking a course, "Abstract Algebra I & Number Theory" and I'm wondering: what is the difference between abstract algebra and number theory? the two topics seem meshed together. i tried googling both of them and it doesn't really help. it's hard to tell the differences between...
  44. L

    Abstract Algebra any help is appreciated

    [b]1. On the set of real numbers, R the following operation is defined: *RxR implies (arrow) R, (x,y) implies (arrow) x*y=2(x+y)-xy-2 Find the neutral element of this operation. [b]3. since we know x*e=x, e*x=x, so i attempted: using e as y, because it would just mean y...
  45. A

    Any suggestions for a book on abstract algebra?

    Hello folks! Do you have any suggestions for a book on abstract algebra? Someone gave me this suggestion Algebra - Michael Artin https://www.amazon.com/dp/0130047635/?tag=pfamazon01-20 however there are some bad (and convincing) reviews on amazon.com about this book (although the...
  46. B

    Courses MIT course number for abstract algebra

    Anybody know the name of the MIT course number for Abstract Algebra . Is it even listed as a course on the MIT opencourseware website?
  47. W

    Solving Abstract Algebra Problem: Proving Isomorphism & Listing Generators

    Can some one help me, how to solve this problem?. Please explain me how is done, been having problem with the subject Let H be the subgroup of GL(2, R) under Matrix multiplication defined by H = {[ 1 n ]}| n E Z } 0 1 Let 0...
  48. E

    Abstract Algebra: Prove Unit question

    Homework Statement Let R be an Integral Domain. Prove that if a,b are elements of R and both a and b are units in R, then prove a*b is a unit of R. Homework Equations a is a unit in R if and only if there exists an element u in R such that au=1=ua where 1 is the identity element of R...
  49. L

    Abstract Algebra: Proving Normal Subgroup and Isomorphisms

    Homework Statement If G1, G2 are two groups and G = G1 times G2 = {(a,b) such that a is an element of G1, b is and element of G2}, where we define (a,b)(c,d) = (ac, bd), (a) Show that N = {(a, e2) such that a is an element of G1}, where e2 is the unit element of G2, is a normal subgroup...
  50. L

    Proving Normality of Homomorphic Image and Subgroup - Abstract Algebra Homework

    Homework Statement If f is a homomorphism of G onto G' and N is a normal subgroup of G, show that f(N) is a normal subgroup of G'. Homework Equations The Attempt at a Solution Once again, I'm completely lost.
Back
Top