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. I

    Can A be a subset of C if it's disjoint from B?

    Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.
  2. G

    When Does (ab)^n Equal (a^n)(b^n) in Ring Theory?

    Homework Statement Let R be a ring and a,b be elements of R. Let m and n be positive integers. Under what conditions is it true that (ab)^n = (a^n)(b^n)? Homework Equations The Attempt at a SolutionWe must show ab = ba. Suppose n = 2. Then (ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = (aa)(bb) =...
  3. T

    Abstract Algebra: Groups and Subgroups

    Homework Statement The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so...
  4. T

    Abstract Algebra: Groups and Subgroups

    The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so I am pretty lost in...
  5. S

    What is the Center of a Clifford Algebra of Order 2^n?

    Homework Statement Show that the center of a Clifford algebra of order 2^n is of order 1 if n is even, and 2 if n is odd Homework Equations the center of an algebra is the subalgebra that commutes with all elements Clifford algebra of 2^n is defined as being spanned by the bases...
  6. A

    Abstract algebra question (math olympiad)

    Let G be a non-cyclic group of order pn where p is a prime number. Prove that G has at least p+3 subgroups. Could anyone offer a solution to this problem?
  7. N

    Do you know of any really good Abstract Algebra websites with lots of examples?

    Hi. I've just failed my first test in my Abstract Algebra course... I'm sure I scored a zero. So... needless to say, I need help. Do you know of any good websites with lots of examples? Or even a really good book with lots of problems? The textbook we're using is 'A First Course in Abstract...
  8. C

    Abstract Algebra - Smallest subgroup of GL(n,R)

    Homework Statement Let A = \left[ \begin{array}{cccc} 1 & 1 \\ 0 & -1 \end{array} \right] Let B = \left[ \begin{array}{cccc} 1 & 2 \\ 0 & -1 \end{array} \right] Find the smallest subgroup G of GL(n,R) that contains A and B. Also, find the smallest subgroup H of G that contains the matrices...
  9. P

    Abstract Algebra Question

    Show that \langle a,b \rangle = \langle a,ab \rangle = \langle a^-1,b^-1 \rangle for all a and b in a group GI am not sure what this question is asking. Does this notation mean that a the cyclic group is generated by a,b and any combination of the two?
  10. The Chaz

    MHB Possibility of a Separate Forum for NT & Abstract Algebra

    1. There should be a separate (sub)forum for NT. ... and one for abstract algebra, for that matter! 2. Show that there are infinitely many n such that both 6n + 1 and 6n - 1 are composite. Without CRT, if possible. My work... let n = 6^{2k}. Then 6n \pm 1 = 6^{2k + 1} \pm 1... Hmm. Having a...
  11. T

    Abstract Algebra - ideals and generators

    Homework Statement a.) Let a=3-8i and b=2+3i. Find x,y ϵ Z[i] such that ax+by=1. b.) Show explicitly that the ideal I=(85,1+13i) \subseteq Z[i] is principle by exhibiting a generator. Homework Equations Given ideal: I=(85,1+13i) \subseteq Z[i] a=3-8i b=2+3i Honestly, I am beyond lost...
  12. G

    Abstract Algebra Proof question

    Homework Statement Let a=p_{1}^{r_{1}}p_{2}^{r_{2}}...p_{k}^{r_{k}}, b=p_{1}^{s_{1}}p_{2}^{s_{2}}...p_{k}^{s_{k}} where p_{1},p_{2},...,p_{k} are distinct positive primes and each r_{i},s_{i} ≥ 0 Prove that (a,b)=p_{1}^{n_{1}}p_{2}^{n_{2}}...p_{k}^{n_{k}} \mbox{ where for each } i...
  13. A

    Abstract Algebra: Parity of a Permutation

    Homework Statement How do I determine the parity of a permutation? I think my reasoning may be faulty. By a theorem, an n-cycle is the product of (n-1) transpositions. For example, a 5 cycle can be written as 4 transpositions. Now say I have a permutation written in cycle notation: (1...
  14. S

    Abstract Algebra is it too much?

    I'm an undergrad math major, and this is my first semester taking upper level math. I'm currently taking Abstract Algebra, and feeling pretty intimidated. I mean, I feel out of the loop, I'm trying hard to understand, but I feel overwhelmed, like maybe it's too much for me. Is it normal to feel...
  15. T

    Abstract Algebra for Physics undergrad

    Hi, I'm doing a Physics undergrad and this semester I have the following courses: Thermodynamics, Quantum Mechanics, Numerical Methods, an Astrophysics course, and a Computational Lab. I've also taken Abstract Algebra which has twice as many lectures as any of these. Add to this the fact that I...
  16. P

    Matrix Representation of Permutations: (1874)(36759)

    Write the following in two row matrix form. (1874)(36759) I have [1 2 3 4 5 6 7 8 9] [8 2 6 1 9 7 4 7 3] my problem is couldn't 7 also go to 5 and have 8 going to 7 and 6 going to 7 so I am sure I am wrong but I am not sure why.
  17. 2

    What are some good sources for learning abstract algebra?

    Does anyone here know any good websites or sources to help me learn and understand abstract ablegra?
  18. X

    Abstract Algebra Problem involving the ideals

    Homework Statement Let f:R→S be a homomorphism of rings. If J is an ideal in S and I={r∈R/f(r)∈J}, prove that I is an ideal in R that contains the kernal of f. Homework Equations The Attempt at a Solution I feel like I have the problem right, but would like to have someone look...
  19. X

    Abstract Algebra Problem involving the order of groups

    Homework Statement Let G be a group with identity e. Let a and b be elements of G with a≠e, b≠e, (a^5)=e, and (aba^-1)=b^2. If b≠e, find the order of b. Homework Equations Maybe the statement if |a|=n and (a^m)=e, then n|m. Other ways of writing (aba^-1)=b^2: ab=(b^2)a...
  20. X

    Abstract Algebra Problem using the division algorithm

    Homework Statement Apply the division algorithm for polynomials to find the quotient and remainder when (x^4)-(2x^3)+(x^2)-x+1 is divided by (2x^2)+x+1 in Z7. Homework Equations The Attempt at a Solution I worked the problem and got that the quotient was (4x^2)-3x-1 and the...
  21. S

    Abstract Algebra Question: order, stabilizer, and general linear groups

    The question asks: 3) Let X be the set of 2-dimensional subspaces of F_{p}^{n}, where n >= 2. (a) Compute the order of X. (b) Compute the stabilizer S in GL_{n}(F_{p}) of the 2-dimensional subspace U = {(x1, x2, 0, . . . , 0) ε F_{p}^{n} | x1, x2 ε F_{p}}. (3) Compute the order of S. (4)...
  22. J

    Abstract Algebra - Properties of Q/Z

    Homework Statement Prove that the group Q/Z under addition cannot be isomorphic to the additive group of a commutative ring with a unit element, where Q is the field of rationals and Z is the ring of integers. Homework Equations The tools available are introductory-level group theory and...
  23. I

    Is Abstract Algebra practical?

    I hear a lot that group theory is important to condensed matter physics. Does it have any practical use? Like if I were to do industry work in materials, would I ever use it? Is it important enough to take a full course on abstract algebra?
  24. L

    Abstract Algebra: Ring Isomorphism Construction

    Homework Statement Homework Equations The Attempt at a Solution
  25. X

    Abstract algebra question concerning center of a group

    Homework Statement If a is the only element of order 2 in a group G, prove that a is an element of Z(G). [Z(G) is the notation used by the book for center of group G] Homework Equations Z(G)={a is an element of G: ag=ga for every g that is an element of G} The Attempt at a...
  26. C

    What are the elements of each order in D_n+Z_9 for n = 7 and 11?

    Pick a number n which is the product of 2 distinct primes 5 or more. Find the number of elements of each order in the groupd D(sub)n+Z(sub)9, completely explaining your work. Verify that these number add up to the order of the group. Ive used 7 and 11 as my primes. So now do I use these...
  27. O

    Abstract algebra , intersection of ideals

    Homework Statement prove that <x^m> intersection <x^n> = <x^LCM(m,n)> Homework Equations The Attempt at a Solution ===> let b be in <x^n> intersection <x^m> then for some t,k,p in Z, b=x^(mt) = x^(nk) thus b=x^(LCM(m,n) * p i.e. b is in <x^LCM(m,n)> <=== let b be...
  28. 2

    Abstract Algebra Question dealing with Constructible Numbers

    Homework Statement Let c=cos(2pi/5). It can be shown that (4c^2)+(2c)-1=0. Use this fact to prove that a 72degree angle is constructible. Homework Equations The Attempt at a Solution I can see that using the equation and what c equals that you get the statement 0=0 and I know...
  29. O

    Practicality of Abstract Algebra

    well the title itself seems to be a paradox, but, What are some applications of abstract algebra (like groups, fields, and rings)? Apparently this determines the symmetry of particles in physics but what are some real-life, money-making application of group theory? (Yes, I money is one of my...
  30. J

    Are H Union K and Z(g) Subgroups in Group Theory?

    Abstract Algebra Questions... I have two problems that I'm a little puzzled by, hopefully someone can shed some light. 1) Show that if H and K are subgroups of the group G, then H U K is closed under inverses. 2) Let G be a group, and let g ε G. Define the centralizer, Z(g) of g in G to...
  31. J

    This abstract algebra problem seems trivially easy. Did I overlook something?

    Homework Statement The problem seems too easy so I suspect that I am overlooking something important. A problem this easy would be completely out of character for my professor...
  32. B

    Abstract Algebra - Cyclic groups

    (This is my first post on PF btw - I posted on this another thread, but I'm not sure if I was supposed to) I was doing some practice problems for my exam next week and I could not figure this out. Homework Statement Suppose a is a group element such that |a^28| = 10 and |a^22| = 20...
  33. B

    Abstract Algebra - Cyclic groups

    1. Problem: Suppose a is a group element such that |a^28| = 10 and |a^22| = 20. Determine |a|. I was doing some practice problems for my exam next week and I could not figure this out. (This is my first post on PF btw) 2. Homework Equations : Let a be element of order n in group and let k...
  34. L

    Orders of Quotient Groups (Abstract Algebra)

    Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...
  35. J

    Abstract Algebra Proof: Groups

    Abstract Algebra Proof: Groups... A few classmates and I need help with some proofs. Our test is in a few days, and we can't seem to figure out these proofs. Problem 1: Show that if G is a finite group, then every element of G is of finite order. Problem 2: Show that Q+ under...
  36. D

    Schools Can I take Abstract Algebra as a High School Student?

    I've read up a little bit about Abstract Algebra and it seems like a really interesting subject. A university near me will offer an intro class in it next semester. Trouble is, the university requires Calc III as a prerequisite for the course. I'm taking AP Calc right now at school, but it...
  37. J

    Using the fact that G is abelian in this abstract algebra problem

    I'll post the problem and my attempt at solution all in one picture: In the red step, I'm using commutative multiplication. Am I allowed to do this? I'm not sure, because the subset of G might not be a subgroup, so I don't know if its necessarily abelian like G is. Or does the fact...
  38. S

    Abstract Algebra: Subgroup Proof

    Homework Statement Show that if H is a subgroup of G and K is a subgroup of H, then K is a subgroup of G. Homework Equations The Attempt at a Solution Well I know that H is a subgroup of G if H is non empty, has multiplication, and his inverses. So I assume that K is a subgroup...
  39. I

    Abstract Algebra - Subgroup of Permutations

    Homework Statement A is a subset of R and G is a set of permutations of A. Show that G is a subgroup of S_A (the group of all permutations of A). Write the table of G. Onto the actual problem: A is the set of all nonzero real numbers. G={e,f,g,h} where e is the identity element...
  40. C

    Abstract Algebra dihedral group

    Homework Statement Let G be a finite group and let x and y be distinct elements of order 2 in G that generate G. Prove that G~=D_2n, where |xy|=n. I have no idea how to solve this or even where to begin. I tried setting up G=<x,y|x^2=y^2=1=(xy)^n> But couldn't get any farther, I am so...
  41. U

    How Can You Prove (ab,c) = 1 Given (a,c) and (b,c) Are Both 1?

    Homework Statement If (a,c) = 1 and (b,c) = 1, prove that (ab,c) = 1. Note that (x,y) refers to the greatest common divisor between x and y. 2. The attempt at a solution There is a theorem that says since (a,c) = 1, there exist integers u and v such that au + cv = 1. Likewise, there...
  42. A

    Abstract Algebra: Quotienting and the First Isomorphism Theorem

    Homework Statement Let T be a subset of S and consider the subset U(T)={f \in A(S) | f(t)\inT for every t\inT}. 1) If S has n elements and T has m elements, how many elements are there in U(T)? 2) Show that there is a mapping F:U(T) -> Sm such that F(fg)=F(f)F(g) for f,g\inU(T) and F is onto...
  43. A

    Does one need to know elementary number theory to study Abstract Algebra?

    It's been some time that I've been studying abstract algebra and now I'm trying to solve baby Herstein's problems, the thing I have noticed is that many of the exercises are related to number theory in someway and solving them needs a previous knowledge or a background of elementary number...
  44. A

    A problem from Herstein's Abstract algebra

    Homework Statement if f \in Sn show that there is some positive integer k, depending on f, such that fk=i. (from baby Herstein). The Attempt at a Solution Suppose that S={x1,x2,...,xn}. Elements of Sn are bijections from S to S. to show that fk=i it's enough to show that fk(xm)=xm for every...
  45. J

    Proving a set under an operation is associative. (Abstract Algebra)

    Homework Statement I'm trying to prove that this is a group. I already established elsewhere that it is a binary operation, so now I am onto proving associativity. I've tried many examples and so I'm confident it is associative, but now I just have to prove that.The Attempt at a Solution...
  46. J

    I'm not sure if this simple first day Abstract Algebra exercise is correct

    Prove: If x has a right inverse given by a and a left inverse given by b, then a = b.The Attempt at a Solution One thing that bothers me: how can we even talk about a left inverse or a right inverse without establishing that x is in an algebraic structure? I wrote this in my proof but I'm not...
  47. M

    Abstract Algebra Proof: gcd(s,t)=r and st=r+v

    Homework Statement Let r,s,t and v be integers with r>0. If st=r+v and gcd(s,t)=r, then gcd(v,t)=r Homework Equations Just stumped. I am not sure what to do next.The Attempt at a Solution There are 2 integers d and e such that S=dR and T=eR, and 2 integers a and b such that Sa+Tb=R. I know I...
  48. N

    [abstract algebra] Isomorphic group of units

    Homework Statement Given that gcd(n,m)=1, prove that \mathbb Z_{nm}^\times = \mathbb Z_n^\times \oplus \mathbb Z_m^\times. Homework Equations / The Attempt at a Solution I can prove both groups have the same amount of elements (using Euler's totient function), but I can't figure out...
  49. N

    Can you recommend me to a book in Abstract Algebra and pre-requistes ?

    Can you recommend me to a book in Abstract Algebra and pre-requistes ?
  50. F

    Proving R[x] is a Principal Ideal Domain Implies R is a Field

    Homework Statement Let R be an integral domain and suppose that R[x] is a principal ideal domain. Show that R is a field. Homework Equations I don't know where to start, I'm not familiar with this material. I was browsing through an abstract algebra book and found this. Would like...
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