What is Abstract: Definition and 532 Discussions

Abstract expressionism is a post–World War II art movement in American painting, developed in New York City in the 1940s. It was the first specifically American movement to achieve international influence and put New York at the center of the Western art world, a role formerly filled by Paris. Although the term "abstract expressionism" was first applied to American art in 1946 by the art critic Robert Coates, it had been first used in Germany in 1919 in the magazine Der Sturm, regarding German Expressionism. In the United States, Alfred Barr was the first to use this term in 1929 in relation to works by Wassily Kandinsky.

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  1. D

    Is Abstract Algebra the Key to Unlocking Mathematical Concepts?

    I have started to write Abstract Algebra notes as I am learning them, and typing them with LaTex afterwards. I have just done a bit but I want some of you to help and see if I have got any thing wrong (having the wrong concept in your mind can have terrible consequences) or anything else to make...
  2. R

    Abstract algebra, left and right translations, what are these good for?

    Homework Statement Does anyone know what left and right translations are good for? \begin{cases} R_{a}g=ga\\L_{a}g=ag\end{cases} with a,g\in G and G is a Lie group How can we interpret these relations in the easiest way like we try to explain it to a student which if not familiar with...
  3. Y

    How to denote tetrad in Abstract Index Notation ?

    I like Penrose's Abstract Index Notation very much. I am familiar with using Abstract Index Notation to denote Coordinate Basis. But when I try to denote tetrad with Abstract Index Notation, I meet problems. How to denote tetrad in Abstract Index Notation?
  4. F

    Exploring Ring Homomorphisms from Z to Z: Homework Equations and Attempts

    Homework Statement number of ring homomorphisms from Z \rightarrow Z? Homework Equations The Attempt at a Solution According to this information on ring homo, There is no ring homomorphism Zn → Z for n > 1. But I guess that doesn't hold for when n = 1, any ideas
  5. N

    Exploring Abstract Algebra: Helpful Links and Practice Problems

    I was wondering if anyone knew any links on the Internet that help to explain abstract algebra and maybe works through some problems as well. Thank you in advance
  6. V

    Abstract Algebra - Polynomials: Irreducibles and Unique Factorization

    Homework Statement Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations Theorem 4.8 Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...
  7. M

    Proof of Aut(G): ϕ(Z(G))= Z(G)

    Homework Statement For every ϕ in Aut(G), ϕ(Z(G))= Z(G). Homework Equations Z(G):={g in G| gh=hg for all h in G} The Attempt at a Solution I haven't made too much progress on this one. I know that if I let g be an element of Z(G) that I need to prove that For every ϕ(g) is also...
  8. M

    Proving |H intersect K| = q for subgroup H and K in G of order pqr.

    Homework Statement Suppose |G| = pqr where p, q, and r are distinct primes. If H is a subgroup of G and K is a subgroup of G with |H| = pq and |K| = qr, then |H intersect K| = q. Homework Equations NA The Attempt at a Solution I have so far: Let a be an element of H intersect K...
  9. M

    Abstract Algebra[zero polynomial of infinite field]

    What would be the best way to show that if F is an infinite field and f(x) is a polynomial in F[x] and f(a)=0 for an infinite number of elements a of F, that f(x) must be the zero polynomial? It kind of just makes logical sense to me, so I can't think of a way to actually show this. please help
  10. N

    Introduction to Abstract Algebra

    I was wondering if anyone could give me any links or an introduction to abstract algebra. I know that abstract algebra is a tough concept to understand (at least for some people, but it varies from person to person). If anyone could help with the basics of it would be greatly appreciated.
  11. M

    Simple Abstract Algebra Problem

    Homework Statement Let G be a nonempty finite set closed under an associative operation such that both the left and right cancellation laws hold. Show that G under this operation is a group. Homework Equations My book defines the left and right cancellation laws as : "For any a,b in...
  12. H

    Abstract Algebra: Proving G is Isomorphic to H with Log

    1. Homework Statement [/b] The set of positive real numbers, R+, is a group under normal multiplication. The set of real numbers, R, is a group under normal addition. For the sake of clarity, we'll call these groups G and H respectively. Prove that G is isomorphic to H under the isomorphism...
  13. D

    Abstract Algebra Hello Experts: Proving Theorems About Ideals and Radicals

    Hello Experts, I can't find the proof of this theorems please help me: Given that there is a commutative ring R and 2 ideals I and J, also given that I is included in J I need to prove 1) radical of I is in radical of J 2) radical of radical of ideal I = radical of ideal I...
  14. A

    How do mathematicians think about abstract algebra?

    Hi Folks. I was hoping to pick the brains of some of the mathematicians and mathematically inclined on this site. I'm very interested in how mathematicians think about abstract objects that don't seem to be grounded in anything concrete. In particular, how do mathematicians think to...
  15. R

    Mathematica Mathematica: Plot using abstract variables instead of set values

    Hi All, Does anyone know of a way to make plots in mathematica in terms of variables? For example, suppose you had a function sin(a*x), and you wanted to plot it but did not want to set a to a specific value, the purpose being to have the graph report multiples of a, not specific numbers...
  16. B

    Abstract Algebra and cyclic subgroups

    Homework Statement from Algebra by Michael Artin, chapter 2, question 5 under section 2(subgroups) An nth root of unity is a complex number z such that z^n =1. Prove that the nth roots of unity form a cyclic subgroup of C^(x) (the complex numbers under multiplication) of order n...
  17. MathWarrior

    Abstract Algebra vs Number Theory?

    I was wondering if one wanted to pursue learning more about cryptography which of these classes would be the most important? Number theory of abstract algebra?
  18. L

    Abstract Algebra Question: Maximal Ideals

    Homework Statement a) Show that there is exactly one maximal ideal in Z_8 and in Z_9. b) Show that Z_10 and Z_15 have more than one maximal ideal. Homework Equations I know a maximal ideal is one that is not contained within any other ideal (except for the ring itself) By...
  19. S

    Abstract Algebra: Properties of the Group U(n)

    Homework Statement (This is an example of a group in my text). An integer 'a' has a multiplicative inverse modulo n iff 'a' and 'n' are relatively prime. So for each n > 1, we define U(n) to be the set of all positive integers less than 'n' and relatively prime to 'n'. Then U(n) is a group...
  20. S

    Abstract Algebra: Question About the Elements in U(n)

    Homework Statement For any integer n>2, show that there are at least two elements in U(n) that satisfy x^2 = 1. Homework Equations None The Attempt at a Solution If the definition of the group U(n) is "the set of all positive integers less than n and relatively prime to n" then the...
  21. S

    Applications of abstract algebra to engineering

    I was wondering if there are any applications of abstract algebra to engineering and where I can go to learn about them?
  22. M

    Abstract Algebra: define an operation

    Homework Statement Does the rule g*x = xg^-1 define an operation of G on G? Homework Equations The Attempt at a Solution I don't even know what this means. Could someone just tell me what it means for a rule to define an operation of one group on itself? I should be able to figure...
  23. K

    Do I need a lot of abstract algebra knowledge to start learning Lie algebra

    I'm a physics undergrad and doing some undergrad study on QFT, and I found that Lie algebra is often invoked in texts, so I decide to take a Lie algebra this sem but I've not taken any abstract algebra course before.The first day's class really beats me because the lecturer used many concepts...
  24. C

    Abstract algebra questions relating to Ideals and cardinality of factor rings

    Homework Statement Find the number of elements in the ring Z_5[x]/I, where I is a) the ideal generated by x^4+4, and b) where I is the ideal generated by x^4+4 and x^2+3x+1. Homework Equations Can't think of any. The Attempt at a Solution I started by finding the zeros of the...
  25. N

    Is Abstract Algebra Worth Taking for Physics?

    Would this be a good thing to take? More specifically, will a introduction to this shed light on/put on more solid ground many of the techniques/organizations in physics that are presented as "tricks"? I just want to be sure it will be worth it, since i'll be taking it alongside...
  26. marcus

    Role of philosophy in QG (not too abstract)

    I assume there is no single right philosophy-of-physics (p-o-p) doctrine. No pretended philosophical "high ground". PoP is an academic subject--there are conferences and workshops. There is a fat two-volume Handbook of Phil. of Phys. published by North Holland Press. Courses are taught, seminars...
  27. K

    Proving Normality of Subgroups in Cyclic Groups

    I'm taking this course "abstract algebra" at university and I've been given some homework questions. I was able to solve all of them but one. And it would be great if anyone could help me with this. The question is like this: "If all cyclic subgroups of G are normal, then show that all...
  28. D

    Abstract Algebra: Groups of Permutations

    Homework Statement List the elements of the cyclic subgroup of S_6 generated by f = \left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 3 & 4 & 1 & 6 & 5\\ \end{array}\right)Homework Equations The Attempt at a Solution I really do not understand what the elements of a permutation really...
  29. M

    Should I take Combinatorics or Abstract Algebra?

    I am a senior student double majoring in computer science and mathematics with the intention of getting a p.h.d in theoretical computer science(either computational complexity or applied discrete mathematics). for the upcoming winter semester I can take 1 math course. The ones that are related...
  30. B

    Determinant proof from abstract algebra

    Homework Statement Let A be a a square n*n matrix. Prove that A^-1 has only integer enteries if and only if the determinant of A is + or -1. Homework Equations general knowledge of determinants The Attempt at a Solution Proof: => Suppose that det(A) = 1 (without losing...
  31. C

    Abstract Algebra First Isomorphsm Theorem

    Homework Statement Use the First Isomorphism Theorem to show that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)} Homework Equations First Isomorphism Theorem: If f: G-> H is a homomorphism then G/ker(f) is isomorphic to im(f) The Attempt at a Solution I understand that I need to show...
  32. T

    Solve Abstract Algebra Homomorphism Problems with Step-by-Step Guidance

    abstract algebra ...HELP, PLZ! THIS IS THE PROBLEM: COMPUTE THE INDICATED QUANTITIES FOR THE GIVEN HOMOMORPHISM KER (PHI) AND PHI(18) FOR PHI: Z -> Z10 (SUBCRIPT) SUCH THAT PHI(1)=6 Can anyone please help me to solve this problem? I don't even know what it's asking for? Don't know where...
  33. D

    Defining Time: Is It an Abstract Concept in Philosophy?

    Is time an abstract object?
  34. Truecrimson

    The best edition of Fraleigh's Abstract Algebra?

    I'm going to buy A First Course in Abstract Algebra by Fraleigh. I've looked at 6th and 7th ed. 6th doesn't have a section on homology groups, but 7th does. From what I found from other threads here, 4th also has homology groups, and 3rd is at least good on group actions. (I haven't got to group...
  35. T

    Are the universe and time abstract concepts?

    It was suggested that this belongs in philosophy so I have moved as I was enjoying the thread. Anyone have any views? ...Is the universe an abstract concept. Is time an abstract concept? Originally Posted by TheAlkemist View Post Yes and yes. An abstract concept = A mental...
  36. B

    Abstract Algebra, Division Ring question

    Homework Statement Let R = { [ a + b*sqrt(m) c + d*sqrt(m) ] } [ n(c - d*sqrt(m)) a - b*sqrt(m) ] (Sorry if the matrix is unclear... I can't get it space nicely. r11 = a + b*sqrt(m) r12 = c + d*sqrt(m) r21 = n(c -d*sqrt(m))...
  37. S

    Solve Abstract Algebra Homework: Finding Left Coset (1,2,3)H with Permutations

    One of my homework problems asks me to list the left coset (1,2,3)H where σ=(1,4,5)(2,3) and H=<σ>. I know that you have to take the do the permutation of (1,2,3)(1,4,5)(2,3) but i am not sure how you can do that? I got (1,2,3)H={(1,2,3)(3)(1,2,4,5)} but i do not think that is right
  38. L

    A More Abstract Definition of an Inner Product Space?

    An inner product space is often simply described as a vector space with the addition of an inner product, but when it comes to the formal definition, the basefield seems to always be restricted to the fields of real and complex numbers. The Wikipedia article on inner product spaces remarks that...
  39. J

    Abstract Algebra, rings, zero divisors, and cartesian product

    The problem states: Let R and S be nonzero rings. Show that R x S contains zero divisors. I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element. R x S is the Cartesian Product so if we have two rings R and S If r1 r2 belong to R and s1 s1...
  40. C

    Abstract algebra-> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with

    Abstract algebra--> Let R be a ring and let M2(R) be the set of 2 x 2 matrices with Homework Statement Let R be a ring and let M2(R) be the set of 2 x 2 matrices with entries in R. De fine a function f by: f(r) = (r 0) <----matrix ...(0 r) for any r ∈ R (a) Show that f is a...
  41. E

    Abstract Algebra: conjugates of cyclical groups

    Homework Statement If G is a group with operation * and \alpha,\beta\in G, then \beta\ast\alpha\ast\beta^{-1} is called a conjugate of G. Compute the number of conjugates of each 3-cycle in S_{n} (n\geq3). Homework Equations The Attempt at a Solution For any group S_{n} there...
  42. N

    How Does Cycle Length Relate to Element Order in Group Representations?

    abstract algebra question?? here is the problem from abstract algebra, anyone could help? Thanks a lot! let G be a finite group. Show that in the disjoint cycle form of the right regular representation Tg(x)=xg of G each cycle has length | g |. (Tg(x) means T sub g of x) loofinf...
  43. K

    Abstract - one to one and onto questions

    Homework Statement Determine whether the given function is one to one and whether it is onto. If the function is both one to one and onto, find the inverse of the function. f:R^{2}\rightarrowR^{2}, f(x,y)=(x+y, y) . Homework Equations The Attempt at a Solution I know one to one...
  44. E

    Why Is Abstract Algebra So Challenging to Visualize Compared to Calculus?

    I am a junior in college right now, and after finishing the Calculus sequence and having my first semester of Analysis, I am now taking Abstract Algebra. I did alright in Analysis but not as good as I had hoped to do. My biggest problems are that, unlike Calculus, which for the most part I...
  45. silvermane

    Proving Greatest Common Divisor of a,b,c | Abstract Algebra 1

    1. The problem statement: Consider 3 positive integers, a, b, c. Let d_{1} = gcd(b,c) = 1. Prove that the greatest number dividing all three of a, b, c is gcd(d_{1},c) 3. My go at the proof and thoughts: Well, I know that the common divisors of a and b are precisely the divisors of...
  46. A

    Abstract algebra vector space problem

    Homework Statement We have a vector space (V, R, +, *) (R being Real numbers, sorry I couldn't get latex work..) with basis V = span( v1,v2). We also have bijection f: R² -> V, such as f(x,y) = x*v1+y*v2. Assume you have inner-product ( . , . ): V x V -> R. ( you can use it abstractly and...
  47. K

    Is Divisibility Sufficient for Proving a Product Divides Another Product?

    Homework Statement Show that if a | c and b | c, and (a, b) = d, then ab | cd.Homework Equations The Attempt at a Solution Abstract divisibility. We have c=am, c=bn, and d=an+bm.
  48. K

    Abstract prime factorization proof

    Homework Statement A positive integer a is called a square if a=n^2 for some n in Z. Show that the integer a>1 is a square iff every exponent in its prime factorization is even. Homework Equations The Attempt at a Solution Well, I know a=p1^a1p2^a2...pn^a^n is the definition of...
  49. C

    Abstract algebra proof involving prime numbers

    The question states prove, If p is prime and p | a^n then p^n | a^n I am pretty sure I have i just may need someone to help clean it up. There are two relevant theorems i have for this. the first says p is prime if and if p has the property that if p | ab then p | a or p | b the...
  50. K

    Abstract - Prove (A-B)union(B-A)=(AunionB)-(AintersectB)

    Homework Statement Prove: (A-B)\cup(B-A)=(A\cupB)-(A\capB) Homework Equations The Attempt at a Solution We need to show (A-B)\cup(B-A)\subseteq(A\cupB)-(A\capB) and (A\cupB)-(A\capB)\supseteq(A-B)\cup(B-A). We begin by showing the first: Let x\in(A-B)\cup(B-A). By...
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