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. R

    Abstract algebra: f(x) is reducible so is f(x+c)

    Homework Statement Let F be a field and f(x) in F[x]. If c in F and f(x+c) is irreducible, prove f(x) is irreducible in F[x]. (Hint: prove the contrapositive) Homework Equations So, I am going to prove if f(x) is reducible then f(x+c) is reducible. The Attempt at a Solution f(x)...
  2. N

    Abstract Algebra - lifting up a factor group

    Abstract Algebra -- lifting up a factor group After spending an extended period with my Professor during office hours I must admit I am mystified. He kept on talking about "lifting up" factor groups. I think this has something to do with using a factor group, say G/N, to show that there...
  3. N

    Abstract Algebra - no Sylow allowed

    Abstract Algebra -- no Sylow allowed Please note Sylow's theorem(s) may not be used. Using Theorem 1 as a tool, prove that if o(G)=p^{n}, p a prime number, then G has a subgroup of order p^m for all 0\leq m\leq n. Theorem 1: If o(G)=p^{n}, p a prime number, then Z(G)\neq (e). Theorem 1 uses...
  4. N

    Abstract Algebra - group

    Abstract Algebra -- group Show that in a group G of order p^2 any normal subgroup of order p must lie in the center of G. I am pretty sure here that p is supposed to be a prime number, as that is the stipulation in preceding and later problems. However, the problem statement does not...
  5. N

    Abstract Algebra - automorphism

    I have two problems I would like to discuss. 1.For any group G prove that the set of inner automorphisms J(G) is a normal subgroup of the set of automorphisms A(G). Let A be an automorphism of G. Let T_{g} be an inner automorphism, i.e. xT_{g}=g^{-1}xg The problem can be reduced to the...
  6. S

    What Group Contains Elements a and b in Abstract Algebra?

    1) find a group that contains elements a and b such that ︱a︱=︱b︱= 2 and a) ︱ab︱ = 3 b) ︱ab︱=4 c) ︱ab︱=5 2) suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40. determine H? does anyone can help me out? and ...i am really in...
  7. N

    Abstract Algebra - isomorphism question

    Abstract Algebra -- isomorphism question If N, M are normal subgroups of G, prove that NM/M is isomorphic to N/N intersect M. That's how the problem reads, although I am not sure how to make the proper upside-down cup intersection symbol appear on this forum. Or how to make the curly "="...
  8. quasar987

    Abstract Algebra: Show E is a Group if Left Identity & Inverse Exist

    Homework Statement I am asked to show that if E is a semi-group and if (i) there is a left identity in E (ii) there is a left inverse to every element of E then, E is a group.The Attempt at a Solution Well I can't seem to find the solution, but it's very easy if one of the two "left" above is...
  9. B

    Abstract algebra question

    Homework Statement Prove that if (ab)^2=a^2*b^2, in a group G, then ab =baHomework Equations No equations necessary for this proofThe Attempt at a Solution Suppose (ab)^2=a^2*b^2. Then (ab)^2=(ab)(ab)=(abba)=(ab^2*a)=a^2 *b^2=> (ab)(ba)=(ba)(ab) = e By cancellation, (ab)=(ba) <=> (ba)=(ab)
  10. B

    Can {1,2,3} and {1,2,3,4} be Groups under Multiplication Modulo 4 and 5?

    Homework Statement Show that {1,2,3} under multiplication modulo 4 is not a group but that {1,2,3,4} under multiplication modulo 5 is a group Homework Equations a mod n=r ;a=qn + r The Attempt at a Solution I'm going to assume when the problem says modulo 4, the problem is read...
  11. T

    Crash course in abstract algebra

    I'm thinking about taking the math GRE in December but I've never studied abstract algebra--all this about rings and groups just flies right over my head. Can anyone recommend a good introductory book? I'm thinking one of the Dover works might be good since they seem to emphasize problem...
  12. T

    Can You Succeed in Abstract Algebra Without Taking Calculus 3?

    I am currently signed up for an intro abstract algebra course. I will be taking this course and calculus 3(stewart's book). I am pretty good at writing proofs. Do you have to know calculus 3 to do well in abstract algebra? Or can you take it concurrently? Is abstract algebra considered a...
  13. F

    Does the Union Notation in Abstract Algebra Allow for Multiple Matches for x?

    This question links to a former discussion on the board. I'm confused regarding this thread: https://www.physicsforums.com/showthread.php?t=3622" Specifically, towards the end of the thread, the asker states (in regards to the union notation originally cited): "...if we say that x is an...
  14. R

    Another Abstract Algebra Question

    Another Abstract Algebra Question... Every symmetry of the cube induces a permutation of the four diagonals connecting the opposite vertices of the cube. This yields a group homomorphism φ from the group G of symmetries of the Cube to S4 (4 is a subscript). Does φ map G onto S4? Is φ 1-1? If...
  15. R

    Abstract Algebra Questions: Homomorphisms and Normal Subgroups

    Abstract Algebra Questions... Help Please! Any and all help on these problems would be greatly appreciated. Thank you in advance to any who offer help :smile:. 1. Let φ:G->H be a group homomorphism, where G has order p, a prime number. show that φ is either one-to-one or maps every element...
  16. C

    Proof: Group Action GxX -> X |X|=|X^G|modp

    G is a finite group, |G| =p^n, p prime *:GxX -> X is group action. X is a finite set, I am required to prove the following |X|\equiv |X^G|modp Now we start by asserting that x_1, x_2, ...,x_m is the set of m orbit representatives. That orbit x <x_i> = {x_i} \\ iff x_i is a...
  17. R

    Abstract Algebra: M(R) 2x2, units, where does the determinant come from?

    Homework Statement Prove that (a b c d) is a unit in the ring M(R) if and only if ad-bc !=0. In this case, verify that its inverse is (d/t -b/t -c/t a/t) where t= ad-bc. Homework Equations An element a in a ring R with identity is called...
  18. R

    How Do Units and Zero Divisors in Rings Relate?

    Homework Statement From An Introduction to Abstract Algebra by T. Hungerford Section 3.2 #29 Let R be a ring with identity and no zero divisors. If ab is a unit in R prove that a and b are units. Homework Equations c is a unit in R if and only if there exists an...
  19. P

    Help with Abstract Algebra: Show ac=b, da=b w/Hint

    Please I need your help for that qustion and how do slove that qustion's problem. can you help me for slove for that? Pleasee Let G be any group, and let a, b ∈ G. Show that there are c, d ∈ G such that ac = b and da = b. Hint: you have to give an explicit definition for c and d in terms of a...
  20. W

    Abstract Algebra group problem.

    ProbelmLet p and q be distinct primes. Suppose that H is a porper subset of the integers and H is a group under addition that contains exactly three elements of the set {p, p+q, pq, p^q, q^p}. Determine which of the following are the three elements in H. a.pq, p^q, q^p b. p+q, pq, p^q c. p...
  21. B

    Abstract Algebra: Splitting Fields and Prime Polynomials

    I'm having trouble understanding splitting fields. Some of the problems are find the degree of the splitting field of x^4 + 1 over the rational numbers and if p is a prime prove that the splitting field over the rationals of the polynomial x^p - 1 is of degree p-1. I'm really confused with these...
  22. H

    Exploring β: A Homomorphism and Isomorphism in Abstract Algebra Assignments

    can anyone help me with my abstract algebra assignment? Let a be an fixed element of some multiplicative group G. Define the map β: Z > G from the additive inter group Z to G by β(n)=a^n. i. Prove that the map β is a homomorphism. ii. Prove/Disprove that the map β is an isomorphism. thanks!
  23. quasar987

    Taking PDE or abstract algebra

    In my uni I am forced to make a painful choice btw taking PDE or abstract algebra. I will take algebra, but I'd like to know what I will be missing? What is being taught in this class exactly? (BESIDES HOW TO SOLVE A PDE BY SEPARATION OF VARIABLES :rolleyes:)
  24. B

    Abstract Algebra - Direct Product Question

    I'm supposed to find a non-trivial group G such that G is isomorphic to G x G. I know G must be infinite, since if G had order n, then G x G would have order n^2. So, after some thought, I came up with the following. Z is isomorphic to Z x Z. My reasoning is similar to the oft-seen proof...
  25. A

    Lagrange's Theorem (Order of a group) Abstract Algebra

    Can someones tells me how to prove these theorems. 1. Prove that if G is a group of order p^2 (p is a prime) and G is not cyclic, then a^p = e (identity element) for each a E(belongs to) G. 2. Prove that if H is a subgroup of G, [G:H]=2, a, b E G, a not E H and b not E H, then ab E H. 3...
  26. R

    Proving Homomorphism and Finding Kernel in Abstract Algebra

    Given R=all non-zero real numbers. I have a mapping Q: R-> R defined by Q(a) = a^4 for a in R. I have to show that Q is a homomorphism from (R, .) to itself and then find kernel of Q. In order to prove homomorphism i did this, for all a, b in R Q(ab) = (ab)^4 = a^4b^4 = Q(a)Q(b). Is...
  27. JasonJo

    Proving H as a Subgroup of S5 | Abstract Algebra Help

    1) prove that H is a subgroup of S5 (the permutation group of 5 elements). every element x in H is of the form x(1)=1 and x(3)=3, meaning x moves 1 to 1 and moves 3 to 3. does your argument work hen 5 is replaced by a number greater than or equal to 3? 2) Let G be a group. prove or disprove...
  28. B

    Abstract Algebra Problem (Group Isomorphisms)

    Hi. My latest question concerns the following. I must prove that the alternating group A_n contains a subgroup that is isomorphic to the symmetric group S_{n-2} for n = 3, 4, ... So far, here's what I have (not much). The cases for n = 3 and n = 4 are elementary, since the group lattices...
  29. E

    Solve Abstract Algebra Problems with Expert Help | Abstract Algebra Assistance"

    Abstract algebra help please! I'm not sure if I've posted in the correct forum but I would like some help with the following question: http://i12.tinypic.com/33mlik6.jpg" I've to complete this table but I am unsure of how to do the very first step which is to fnd wv. I am learning this...
  30. H

    Urgend Assistance needed: Abstract Algebra

    Hi I'm fairly new at abstract algebra and have therefore got stuck with this assignment. Hope there is somebody here who can help me complete it, because I have been ill these last couple of weeks. Its goes something like this b is a number written in base 10 b\;= \;b_010^0 +...
  31. M

    Understanding Abstract Algebra to Groups, Modules, and Representations

    I have finally understood algebra. It is all about understanding groups. first of all we tackle abelian groups, and finitely generated ones at that., we completely classify them as sums of cyclic groups using their structure as Z modules where Z is the integers. then we ask about non...
  32. L

    Another question from Elements of Abstract Algebra by Clark - transformation groups

    This is question 53\gamma. Given a group G of transformations that acts on X... and a subgroup of G, Go (g * x = x for all x for each g in Go), show that the quotient group G/Go acts effectively on X. A group G "acts effectively" on X, if g * x = x for all x implies that g = e, where g is a...
  33. U

    Answer Abstract Algebra Questions - LCM & Subgroups

    hello i have two questions and i need answers for them first one: in the additive group (Z,+) show that nZ intersection mZ= lZ , where l is the least common multiple of m and n. The second question is : Given H and K two subgroups of a group G , show the following...
  34. S

    Help Needed: Abstract Algebra Textbooks & Dual/Quotient Vector Spaces

    Lately I've been taking a unit that deals with abstract algebra and I'm finding myself not understanding the lectures at all. To make matters worse the unit doesn't have a reccomended textbook so I don't even have any infomation to self learn from. I guess what I'm asking is for some good...
  35. S

    Few problems related to abstract algebra

    Hi everyone; There are some questions which are frizzling my mind, if anybody could help then please reply to these ques which are as follows. Q1) Prove that homomorphic image of cyclic group is itself cyclic? Q2) Prove that any group 'G' can be embedded in a group of bijective mapping of...
  36. MathematicalPhysicist

    Ancient puzzle, solved by abstract algebra?

    i reackon youv'e seen it already, the problem is to rearrange the next numbers in the fixed order: 1 2 3 4 5 6 7 8 9 11 10 when you have at the last entry a vacant place you need to put it in order. this is from the text of edwin h. connell, and i think it's impossible (after a lot...
  37. E

    Questions concerning Finite Fields (Basic Abstract Algebra)

    Hello, everyone, i am a newbie here. I am currently taking a modern linear algebra course that also focus on vector spaces over the fields of Zp and complex numbers. Since i am not familiar with typing up mathematics using tex or anything so that i can post on the forums, i will use the...
  38. H

    Abstract Algebra: Applications & Overview

    I just started with the course of discrete mathematics,,where we have abstract algebra..I am actually interested in the application of this algebra..i know that this is used in Cryptography,error correcting codes,and theoritical computer science...i just want to have a basic outline of how they...
  39. JasonRox

    Looking for an Advanced Abstract Algebra Text

    Does anyone know a good Abstract Algebra text? I currently have the text by Gallian and quite frankly I give two stars out of five. I'm looking for something that's more advanced, and well written. Any recommendations will be greatly appreciated. :biggrin:
  40. mattmns

    Abstract Algebra (Normal Subgroups)

    Hello. We got a review today in abstract algebra, and I am stuck on two problems. 1) Let f: G -> H be a surjective homomorphism of groups. Prove that if K is a normal subgroup of G, then f(K) is a normal subgroup of H. Where f(K)= {f(k): k \inK} The entire f(K) part is really throwing me...
  41. R

    What exactly is Abstract Algebra?

    I'm 17 and my high school has no other math courses to offer me. At a local college there is a course called "Modern Algebra" and I was wondering if it was the same thing as abstract algebra. I asked my math teacher about it, and he said it was the hardest math class he took; he used to call...
  42. F

    Finding the Order of SL(2,Fp) in Abstract Algebra

    Help with abstract algebra! Here is my quetion. What is the order of the set of all 2x2 matricies (such that its entries a,b,c,d are between 0 and p-1), and whose determinant is congruent to 1 modulo p ? => Order of SL(2,Fp) thanks :)
  43. T

    Abstract Algebra Proof needed

    Hello. I was reading a journal and an interesting problem came up. I believe the journal was in the American Mathematics Society publications. Well, here's the statement. "For all integers, n greater than or equal to 3, the number of compositions of n into relatively prime parts is a...
  44. T

    Abstract Algebra - Compositions

    Hello. I was reading a journal and an interesting problem came up. I believe the journal was in the American Mathematics Society publications. Well, here's the statement. "For all integers, n greater than or equal to 3, the number of compositions of n into relatively prime parts is a...
  45. C

    Abstract algebra proof: composition of mappings

    define right-inverse of a mapping B to be mapping A, such that B * A= identity (iota). Where the operation * is composition. Note that B is A's left-inverse. QUESTION: Assume S is a nonempty set and that A is an element of M(S) -the set of all mappings S->S. a) Prove A has a left...
  46. A

    Analysis or Abstract Algebra: Which is More Interesting and Worthwhile?

    This coming fall semester, I have a choice between taking Analysis and Abstract Algebra. Unfortunatly, I'm having a great deal of trouble deciding which to take. Both seem interesting (though Algebra more so). On the other hand, Analysis would open more options for spring semester (in...
  47. I

    Linear and Abstract Algebra textbooks

    I need some recommendations for a good linear algebra textbook, something that's actually used in schools. I've finished linear 1 and 2 and I'm doing some preparation during the summer.
  48. V

    Abstract Algebra: Groups of order 21

    I was given a problem to prove there are at most 3 groups of order 21, with extra credit for proving there are at most 2. I am pretty stuck on this one but here is what I have so far: Suppose G is a group of order 21 Let K be a sylow 3-subgroup of G and let H be a sylow 7-subgroup of G...
  49. V

    Abstract algebra problem concern

    My prof. assigned this problem as the only one for HW a few days back, and for some reason the answer seems too obvious. What subtleties could I possibly be missing? Let G be a group of permutations in a set S. If x, y \in S and y \in orb_g(x), then orb_g(y) = orb_g(x) Still, I am...
  50. Oxymoron

    Determining Quotient Group (\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle

    Question 1 Determine the quotient group (\mathbb{Z}_2\times\mathbb{Z}_4)/\langle(1,2)\rangle Answer \langle(1,2)\rangle is a cyclic subgroup H of \mathbb{Z}_2\times\mathbb{Z}_4 generated by (1,2). Thus H=\{(0,0),(1,2)\} Since \mathbb{Z}_2\times\mathbb{Z}_4 has 2.4 = 8 elements, and...
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