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. T. Wentling

    What background fits promising areas of mathematical physics

    I'm a graduate mathematics student and I did my undergrad in applied math. I also took the normal 10 hrs of physics foundations and then a semester of modern physics (basic quantum intro, special relativity, orbit states etc.). I was thinking about pursuing study in areas that would be...
  2. A

    How to show G/Z(R(G)) is isomorphic to Aut(R(G))?

    I am working on this problem with lots and lots of nesting definitions like this following, and I have been trying to get help from here as well as http://www.quora.com/How-do-I-prove-G-Z-R-G-is-isomorphic-to-Aut-R-G , but none gave me complete help: Show that ##G/Z(R(G))## is isomorphic to a...
  3. A

    Solving Simple Group Homomorphism Problem: Proving phi(G) is Subgroup of N

    I have this problem on simple group's homomorphism: Let ##G′## be a group and let ##\phi## be a homomorphism from ##G## to ##G′##. Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G′## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subset N##. And last year somebody...
  4. AXidenT

    Functional Analaysis or Abstract Algebra or Fields?

    Entering my third year of my bachelor of science majoring in maths/physics and having some trouble deciding what courses to do this semester. I know for sure I will be taking complex analysis and 3rd year quantum however am having trouble picking between 3 in particular for my final two courses...
  5. A

    Validating Logic in a Group Theory Problem

    To make a very long story short, in a group theory problem I am working on, I need to prove this: ##A \lhd B \Rightarrow A'\neq A##, where ##A## and ##B## are finite and ##A'## is called the commutator subgroup: ##\begin{align} A' :&= [A, A] \\ &= \langle [x, y] \mid x, y \in A \rangle \\ &=...
  6. neosoul

    Programs Physics Major: Should I Take Abstract Algebra?

    Should I take abstract algebra. I was going to double major but I don't want to be at school for more than four years or pay for extra classes. Therefore, I decided minor in mathematics instead. I registered for abstract algebra before I decided to just minor in mathematics. I have a hard time...
  7. A

    Automorphism Group of Radical of Finite Group

    I am working on a problem on automorphism group of radical of finite group like this one: Here are what I know and what I don't know: ##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the...
  8. A

    Subnormal p-Sylow Subgroup of Finite Group

    I am self-studying a class note on finite group and come across a problem like this: PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).## Where ##O_p(G)## is the subgroup generated by all subnormal p-subgroups of ##G##; ##E(G)## is the...
  9. A

    Solving Simple Group Problem: Subset of Normal Subgroup of Index 2

    I am working on myself on a problem looks like this: Let ##G'## be a group and let ##\phi## be a homomorphism from ##G## to ##G'.## Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G'## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subseteq N##. I have been asking...
  10. R

    Group Homomorphism & Group Order

    I came across this problem in class note but I was stuck: Assume that ##G## be a group of order 21, assume also that ##G'## is a group of order 35, and let ##\phi## be a homomorphism from ##G## to ##G.'## Assume that ##G## does not have a normal subgroup of order 3. Show that ##\phi (g) = 1##...
  11. T

    Abstract Linear Algebra: Eigenvalues & Eigenvectors

    Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...
  12. T

    Unique Vector a in V such that L(x) = <a,x>

    Homework Statement Let V be a finite-dimensional real vector space with inner product <⋅,⋅> and L: V → R a linear transformation. Show that there exists a unique vector a ∈ V such that L(x) = <a,x>. Homework Equations Hey everyone, so I'm a physics student who had to choose a few electives in...
  13. TheBiologist

    Linear and Abstract Algebra: What Is It?

    Not quite sure, could someone kindly explain the concept of the topic to me? Thanks, it really means a lot. :)
  14. C

    Shoud I take Ring/Field Theory or Complex Analysis?

    Having just finished an introductory course on group theory (with some bits of ring and field theory), I am completely enthralled with this type of math. I initially planned on taking Complex Analysis next semester since so many people say it's "useful" for physics (this was also a compromise...
  15. A

    Talking points in Commutative Algebra, please

    < Mentor Note -- thread moved to HH from the technical math forums > My final assignment in graduate algebra is to write an essay about the relationship among the subjects we have learned so far this semester: (1) Module (2) The Field of Fractions of an Integral Domain (3) Integrality (4)...
  16. S

    Math Elective Help: Abstract Algebra, Theory of Numbers, or Symbolic Logic?

    Hello, I'm debating between taking either abstract algebra, theory of numbers, or intermediate symbolic logic as a math elective. Does anyone have any idea which would make my life easier?
  17. A

    Prime Ideal & Noetherian Integral Domain

    I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for pointers. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, thank you for...
  18. A

    Prime Ideal & Noetherian Integral Domain

    I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for help. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, and especially...
  19. PsychonautQQ

    Low Level Abstract Algebra Question

    Homework Statement Let ab=a and ba=b, show that a^2 = a and that b^2 = b Homework Equations none The Attempt at a Solution Not sure if I did this correct.. but here is what I did. Given: ab = a. Multiply both by left hand multiplication by a^-1 a^-1*a*b = 1. where a^-1*a is obviously...
  20. Euge

    MHB Can a Group Have a Trivial Automorphism Group with Less than Three Elements?

    Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.
  21. W

    Properties of Kernels of Homeomorphisms

    Hi, let ##h: A \rightarrow A ##be a homomorphism between algebraic structures. Is there a nice result describing the properties of ##Ker h^2 ## , where ##h^2 = hoh ## (composition) ? Clearly , ## ker( h) \subset ker (h^2 )## , but are there some other results relating the two; maybe relating...
  22. A

    Abstract Algebra: Beyond and Higher?

    I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.
  23. A

    MHB What is the name of this theorem in Abstract Algebra

    Hi, There is a theorem in Abstract which said if g.c.d(x,y)= d (g.c.d the greatest common divisor between x and y) then there exist an integers a,b such that ax + by = d It is a corollary from Euclidean algorithm. Does it has a name ? Thanks in advance.
  24. Jarvis323

    Should I take Number Theory or Abstract Algebra

    Which course do you think is more important or interesting to take for someone interested in theoretical computer science or theoretical mathematics, number theory or abstract algebra? I am mainly interested acquiring skills and knowledge that will enable me to prove something significant...
  25. D

    MHB Abstract Algebra Sylow Subgroup

    I have a question about abstract algebra so if someone could help me answering this question please ... Suppose P,P' are 3-Sylow subgroup, and let Q be their intersection and N the normalizer of Q. Problem: Explain why is the order of N divisible by 9 ? Thanks for your help. Regards,
  26. Q

    Abstract Algebra: x^p-a irreducible using automorphisms

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  27. A

    Abstract Algebra: Abelian group order

    Homework Statement Let G be an abelian group and let x, y be elements in G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). Assume in addition that (o(x),(o(y)) = 1. Prove that o(xy) = o(x)o(y). The Attempt at a...
  28. H

    Abstract Algebra: need a review of 1-1 and onto proof

    Homework Statement define a function f:H--> gHg^{-1} Homework Equations prove if f is 1-1 and onto.The Attempt at a Solution 1-1: f(h1)=f(h2) gh1g^{-1}=gh2g^{-1} h1=h2 (left and right cancellations) onto: f(g^{-1}hg)=gg^{-1}hgg^{-1}=h so every h belonging to H has an image of g^{-1}hg...
  29. H

    Abstract Algebra: Permutations and Disjoint Cycles

    Homework Statement Theorem 8.1 of Dan Saracino: Let f ε S_{n}. Then there exist disjoint cycles f_{1},f_{2} .. in S such that f= f_{1}°f_{2}... In proving this theorem, it considers a finite group S_n={1,2,..,n} and chooses x_1 ε S_n. Then it defines x_2= f(x_1), x_3=f(x_2) and so on. The...
  30. saybrook1

    Abstract algebra or set theory?

    I'm trying to round out my math skills in order to apply to graduate school for physics and I've already taken all of the calculus offered along with linear algebra, power series etc... I'm wondering which would be better should I choose to take a math course this term: abstract algebra or set...
  31. J

    Abstract Algebra - Isomorphism

    1. Show that S42 contains multiple subgroups that are isomorphic to S41. Choose one such subgroup H and find σ1,...,σ42 such that How can you solve this?? I am confused if anyone can help me to solve this!
  32. T

    Abstract algebra- isomorphisms

    Homework Statement Let A=C_{p^k} where p is a prime and k>0. Let _{p^m} A consist of all element a of A such that a^{p^m}=e. Prove that _{p^m} A/_{p^m-1} A\cong C_p if m\leq k, \frac{_{p^m} A}{_{p^m-1} A}=e if m>kThe Attempt at a Solution Please could someone explain how to get started with...
  33. P

    Abstract Algebra: Prove two kernels are the same

    Homework Statement Prove that (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is isomorphic to F_4[z]/(z^2 + z + 1) by showing that the kernel of \phi : (\mathbb{Z}/2\mathbb{Z})[x,y] \to (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is the...
  34. I

    Abstract Algebra Proof by induction problem

    Homework Statement Show via induction that the nth root of (a1 * a2 * a3 * ... an) ≤ 1/ (n) * ∑ ai, where i ranges from 1 to n. Homework Equations Induction The Attempt at a Solution Let Pn be the statement above. It is clear that P1 holds since a1 ≤ a1. Now let us assume that Pn...
  35. Z

    Abstract Algebra: Solving with Cosets

    Homework Statement Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G. I know that fg is a well-defined permutation. However, we have not established (yet) that G/H is a group. 2 parts to the...
  36. H

    Abstract algebra: elements of fiber writable as

    Greetings, For a homomorphism \varphi, I'm trying to show that elements of a fiber, say the fiber above a, X_a, are writable as a given element of X_a times an element of the kernel K. So, if a\in X_a and b\in X_a, then \exists k\in K such that b=ak. I want to do this without using the...
  37. U

    Abstract algebra, show that phi is a homomorphism

    Homework Statement The Attempt at a Solution I'm very new to this kind of maths, so don't quite know how to get started. If I understood the question at all we have g_i \mapsto \phi_i and so I have a homomorphism if I can show that \pi(g \cdot g_i) = \pi(g) \circ \pi(g_i) I'm thinking...
  38. R

    Abstract Algebra Proof (Cyclic cycles & order)

    Prove that if G is a group and aεG, then o(a-1)=o(a) This is all I have so far: Assume G is a group and aεG. Because G is a group a has an inverse in the group, a-1 s.t. aa-1=e, which is also in G. <a>={an|nεZ}. |<a>| is the number of elements in <a> before it cycles back. Basically all I've...
  39. Schild'sLadder

    Abstract Algebra: book rankings.

    Could someone try to rank 'Abstract Algebra' textbooks, either undergraduate, or graduate level: By how rigorous they are, how they transfer to applicable subjects, and how well they're laid out, in a pedagogical manner. Any answers would be appreciated. Thanks in advance! SL!
  40. S

    Proving something is commutative in abstract algebra

    If \ast : (f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d}), show that \ast is commutative. Note that d|n says d divides n. Now I was not sure how to do this from an abstract algebra point of view although when I stare at it my though process was to maybe rewrite it somehow, which will then be...
  41. alyafey22

    MHB Abstract algebra recommendation

    I am reading at the moment about abstract algebra. It is a very interesting field. I was amazed by the number of examples, applications and related concepts. Never seen something similar in any other mathematical field. I saw lots and lots of theorems and I was wondering whether I should...
  42. stripes

    More intro abstract algebra problems

    Homework Statement Define the set Q[√2] to be the set {a + b√2 | a, b are rationals}, and define addition and multiplication as "usual" (so 2×4 = 8, 2 + 4 = 6, you know, the usual). Show that for any nonzero A in the set Q[√2], there exists an inverse element so that A×A-1 = 1Q[√2]. There...
  43. stripes

    Intro abstract algebra along with basic set theory

    Homework Statement An interesting example of a ring: Begin with a nonempty set X and form the power set of X, P(X), which is the set of all subsets of X. On P(X), define addition + and multiplication × as follows: For A, B in P(X): A × B = A ∩ B A + B = (A\B) ∪ (B\A), where as...
  44. J

    Abstract Algebra or Topology: Which is the Better Choice for a Math Major?

    Hi there, Need one upper div math class to fill out my schedule. It looks like it's a choice between intro to abstract algebra or intro to topology. Which would benefit me more, as a student looking towards grad school?
  45. R

    Abstract Algebra: Relations; Find a symmetric and transitive relation in Z x Z

    Abstract Algebra: Relations; Find a relation that is symmetric, etc Homework Statement Find a relation that is symmetric and transitive but not reflexive. Homework Equations None, other than my chosen condition on the relation, namely: xy > |x + y|. The Attempt at a Solution...
  46. K

    Abstract Algebra Proof Concerns

    Hello. I started Gallians Contemporary Abstract Algebra today. Is it wise to go through each of the given proofs for all of the theorems. For example I just studied the proof for division algorithm. Took Quite some time. I don't know if I could have produced this proof without peeking at the...
  47. A

    Which abstract algebra textbook is most cummulative

    If I were to use an abstract algebra book for quick and easy reference which one would it be? Dummit and Foote is very cumulative, is there anything better in the market? And how long would it take to work out all of D + F for an average student with basic background in Algebra?
  48. N

    Abstract Algebra: Unnecessary Information in D&F Problem Statement?

    Homework Statement Problem 35, Section 7.3 of Dummit and Foote: Let I, J, and K be ideals of R. (a) Prove that I(J+K) = IJ+IK and IJ+IK = I(J+K). (b) Prove that if J \subseteq I then I \cap (J + K) = J + (I \cap K). 2. Concern/Question Despite the problem statement specifically...
  49. K

    [Abstract Algebra] Permutations and shuffling cards

    It's been a while since I've posted. This is a problem I had for a homework assignment a few weeks ago but I completely figure out. Any help appreciated. Homework Statement "A card-shuffling machine always rearranges cards in the same way relative to the order in which they were given to...
  50. T

    Scheduling: Abstract algebra, numerical analysis, Probability, or?

    I need to choose one more math class to reach a full-time status for next fall. So far I am already taking Classical Mech I from Physics Dept, Analysis I and PDE from Math Dept. I hear Analysis is already time-consuming hard class and I guess PDE isn't easy either, so I am considering to...
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