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. J

    Abstract Math, Tautology

    Homework Statement Find a statement for , S, equibalten to ~(P v Q) and show that it is logically equivalent by construction the truth table for "S if and only if ~(P v Q)" and showing that this statement form is a tautology. Homework Equations The Attempt at a Solution My...
  2. G

    Abstract Algebra- A simple problem with Cosets

    I need to find all the cosets of the subgroup H={ [0], [4], [8] ,[12] } in the group Z_16 and find the index of [Z16 : H]. Help would be appreciated :)
  3. O

    Abstract Algebra: Homomorphism f Determined by f(1) in Z

    Homework Statement Let R be any ring and f:Z→R a homomorphism. a)Show that f is completely determined by the single value f(1) b)Determine all possible homomorphisms f in the case when R = Z. Homework Equations The Attempt at a Solution This question has me totally confused...
  4. E

    Abstract Algebra- homomorphisms and Isomorphisms, proving not cyclic

    1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic. 2. Homework Equations - the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K...
  5. S

    Proof: Characteristic of Commutative Ring R[x] is Same as R

    Let R be a commutative ring. Show that the characteristic or R[x] is the same as the characteristic of R. I'm really not sure where to start on this at all. I'm not sure what is ment by R.
  6. S

    Proof: a^3 divides b^2 implies a divides b in Abstract Algebra.

    Let a, b be integers a,b>0 show that if a^3 | b^2 then a|b (Consider the prime factorization of a and b) I've tried setting up generic prime factorization of a and b but then don't get any where, I'm not very strong at this subject. Any kind of hints / where to start would help a lot...
  7. C

    How many groups are there of order 7?

    How many groups are there of order 7. I just need to know this to continue with an assignment. Thanks.
  8. S

    Proving the Commutativity of a Group with Abstract Algebra

    Let G be a group. Show (xy)^{-1} = x^{-1}y^{-1} for all x, g \in G if and only if G is abelian. Homework Equations The Attempt at a Solution
  9. O

    Abstract Algebra: Polynomials problem

    Homework Statement Let f(x)=x5-x2-1 \in C and x1,...,x5 are the roots of f over C. Find the value of the symmetric function: (2x1-x14).(2x2-x24)...(2x5-x54) Homework Equations I think, that I have to use the Viete's formulas and Newton's Binomial Theorem. The Attempt at a...
  10. P

    Any good book to review abstract algebra?

    Hi, I am a junior and a math major, and I am almost done with my year-long abstract algebra sequence for undergraduates. While I found the materials interesting, I feel like I got lost at some places in this course, and I would like to review (or in some topics, relearn) the materials that I...
  11. T

    Proving the Center of a Group Generated by x and y is {e, x^n}

    Homework Statement Let G=<x, y| x^{2n}=e, x^n=y^2, xy=yx^{-1}>. Show Z(G)={e, x^n}. Homework Equations The Attempt at a Solution So I tried breaking this up into cases: Case 1: If n=1. then |x|=1 or 2. If |x|=1, then x=e and x would obviously be in the center. If |x|=2, then xy=yx (since...
  12. T

    Abstract Algebra: Find Generators & Relations for Z2+Z2+Z2

    Homework Statement What is the minimum number of generators needed for Z2+Z2+Z2? Find a set of generators and relations for this group. Homework Equations The Attempt at a Solution I think it is obvious that the minimum amount of generators that you need is three, with Z2+Z2+Z2 =...
  13. M

    Understanding in abstract algebra

    In algebra, do you just base your understanding off the pure definitions and groups? I am learning some multilinear algebra, seeing a lot of talk about rings, algebras, modules, etc. and I can't help but thinking it's all just frivolous, pointless definitions. That's partly because I just can...
  14. K

    Find d when d|n2+n-2, d|n3+2n-1 & d=1 (mod 2), d > 1

    Let d=GCD(n2+n-2,n3+2n-1). Find d if d=1(mod 2) & d > 1. So we know d|n2+n-2 & d|n3+2n-1. My question is simply this, the professor wrote down hence d|n3+n2-2n, right after what is written above. But I'm just not seeing how you get that combination. I understand how to work the...
  15. T

    Abstract Algebra Homomorphism Proof

    Homework Statement Let G and H be two groups. If f: G \rightarrow H is a homomorphism, a \in G and b = f(a). If ord(a) = n, ord(b) = m, then n is a multiple of m. (Let e_{1} be the identity of G and e_{2} be the identity of H) I have to prove that n is a multiple of m. Homework Equations...
  16. T

    Abstract Algebra question

    Homework Statement (1)To prove this I have to let G be a group, with |G|=p^2. (2)Use the G/Z(G) theorem to show G must be Abelian. (3) Use the Fundamental Theorem of Finite Abelian Groups to find all the possible isomorphism types for G. Homework Equations Z(G) = the center of G (a is...
  17. T

    Abstract Algebra: Finding Conjugates

    Homework Statement Consider this group of six matrices: Let G = {I, A, B, C, D, K}, Matrix Multiplication> I =\begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} A =\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix} B =\begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix} C =\begin{bmatrix}-1 & -1\\0 & 1\end{bmatrix} D...
  18. I

    Can (insert topic) be understood through a mathematical lens?

    For everyone out there that has every pulled their hair out writing an abstract for some experimental work, I have a nice recipe for how the first couple sentences of one should go. We have measured (insert type of data i.e. time resolved absorption spectra, x-ray diffraction, surface...
  19. H

    Should I take complex analysis or abstract algebra?

    Being a high school student who will be going into physics, should I take complex analysis or abstract algebra in the fall? I can't take both at once, and I am set to take intro to QM (I will already have taken Calc I-III, an introductory functional analysis course, and linear algebra. I also...
  20. O

    Abstract Alg- Group theory and isomorphic sets.

    Homework Statement I am suppose to determine if the following list of groups are isomorphic and if they are define an isomorphic function for them. a. [5Z, +],[12Z, +] where nZ = {nz | z\inZ} b. [Z6, +6]], [S6, \circ] c. [Z2, +2]], [S2, \circ] Homework Equations +6 means x +6] y = the...
  21. K

    Abstract Algebra, Euclidean Algorithm

    Use the Euclidean Algorithm to find the gcd of the given polynomials: (x3-ix2+4x-4i)/(x2+1) in C[x] First I got x-i R: 3x-3i, then I took the 3x-3i into x2+1 & got 1/3 x R: 1+i. Then I was going to take 1+i into 3x-3i. However that never ends it seems, unless I just confused myself...
  22. C

    Is R={0, 2, 4, 6, 8} a Field under Addition and Multiplication Modulo 10?

    Homework Statement Let R={0, 2, 4, 6, 8} under addition and multiplication modulo 10. Prove that R is a field. Homework Equations A field is a commutative ring with unity in which every nonzero element is a unit. The Attempt at a Solution I know that the unity of R is 6, and that...
  23. K

    Binary Operation, Abstract Algebra

    Define a binary operation on Z, the set of integers by the equation m * n = m + n + mn. Which of the following statemnts is / are true about the binary structure of Z with * 1) * is not associative 2) There is no element e belonging to Z such that for every z belonging to Z, z*e = e*z = z 3)...
  24. E

    Abstract Algebra: Show that 2Z + 5Z = Z

    Homework Statement Show that 2\mathbb{Z} + 5\mathbb{Z} = \mathbb{Z}Homework Equations where 2Z + 5Z = {a+b | a in 2Z and b in 5Z} = ZThe Attempt at a Solution For any n in Z, we can write n= (5-4)n = 5n +(-4)n = 5n + 2(-2n) And since 5n is in 5Z and 2(-2n) is in 2Z, we can form Z from any...
  25. K

    Simple Abstract Proof, with Matrices

    Prove that the field R of real numbers is isomorphic to the ring of all 2 X 2 matrices of the form (0,0)(0,a), with a as an element of R. (Hint: Consider the function f given by f(a)=(0,0)(0,a).) I have no problem showing that it is a homomorphism & that it's injective. My question arrises...
  26. T

    Abstract Algebra: Proving/Disproving |a|=|b| if |a^2|=|b^2|

    Homework Statement If |a^2|=|b^2|, prove or disprove that |a|=|b|. Homework Equations The hint I was given is that let a be an element of order 4n+2 and let the order of b=a2 The Attempt at a Solution I can disprove this by looking at examples, such as in the group Z20 with...
  27. T

    Abstract algebra cyclic subgroups

    Homework Statement Suppose that G is a group with exactly eight elements of order 10. How many cyclic subgroups of order 10 does G have? Homework Equations The Attempt at a Solution I really don't have a clue how to solve this, any help would be greatly appreciated.
  28. F

    Abstract Linear Algebra: Dual Basis

    Homework Statement Define a non-zero linear functional y on C^2 such that if x1=(1,1,1) and x2=(1,1,-1), then [x1,y]=[x2,y]=0. Homework Equations N/A The Attempt at a Solution Le X = {x1,x2,...,xn} be a basis in C3 whose first m elements are in M (and form a basis in M). Let X' be...
  29. icystrike

    Abstract Algebra: Schwarz Inequality Homework

    Homework Statement This is part of the proof of Schwarz inequity. Please help me understand the following equation , i think it should not be a equal sign instead it should be greater or equal to. Homework Equations The Attempt at a Solution
  30. I

    Abstract Algebra: a problem about ideal

    Homework Statement Let J be the set of all polynomials with zero constant term in Z[x]. (Z=integers) a.) Show that J is the principal ideal (x) in Z[x]. b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each n\inZ. Homework Equations The Attempt at...
  31. P

    Help with Plane Lattices Problems in Abstract Algebra

    I have been struggling through this Abstract Algebra class and have completely bogged down in the Wallpaper Patterns chapter, especially the plane lattices section. Can anyone give me some help for the following three problems? I am not sure how to start any of the three problems. Thanks for any...
  32. T

    Is C(a) isomorphic to C(gag-1) for elements a and g in a group?

    Homework Statement If a and g are elements of a group, prove that C(a) is isomorphic to C(gag-1) Homework Equations I have defined to mapping to be f:C(gag-1) to C(a) with f(h)=g-1hg. I have no idea if this is right. The Attempt at a Solution I don't have a clue at the solution...
  33. M

    Abstract Algebra: Solving Stumping Questions | αη = β and G is Abelian

    I have 2 algebra questions which are stumping me, I just can't seem to use my notes to figure them out! 1. Let α, β ∈ S17 where α = (17 2)(1 2 15 17 ), β = (2 3 16)(6 16 17 ). Determine η, as a product of disjoint cycles, where αη = β. 2. Let G be a group in which a^2 = 1 for all a ∈ G...
  34. T

    Abstract Algebra: Proving whether H is a subgroup.

    Homework Statement Let R = {all real numbers}. Then <R,+> is a group. (+ is regular addition) Let H = {a|a \epsilon R and a2 is rational}. Is H closed with respect to the operation? Is H closed with respect to the inverse? Is H a subgroup of G? Homework Equations N/A The Attempt at a...
  35. C

    Abstract algebra. proving things about U(n)

    Homework Statement prove that if g is in Z*_n then g^2=1, so g has order 2 or is the identity. show that the largest value of n for which every non identity element of Z*_n has order 2. which are these others. Homework Equations Z*_n = U(n) different notation it is the the group of co...
  36. J

    Help with abstract algebra proof

    prove that a finite ring with identity has characteristic n for some n>0. been trying for a while getting nowhere any ideas?
  37. T

    Abstract Algebra-Pre Exam questions

    Homework Statement Question 1: Let \sigma_{1}, \sigma_{2} be two cycles of length n in the symmetry group S_{n} . Prove that if \sigma_{1} & \sigma_{2} commute then there is a natural number r such as \sigma_{1}^{r} = \sigma_{2} . Question 2: A. Does the groups D_{3}xZ_{5} and...
  38. T

    What are the subgroups of D12 and how can they be proven to be isomorphic to Dm?

    Homework Statement 1. Let Dn be the dihedral group of order 2n, n>2 . A. Prove that each non-commutative sub-group of Dn isomorphic to Dm for some m. B. Who are all the non-commutative subgroups of D12? 2. Let G be the group of all the matrices from the form: 1 a c 0 1 b 0 0...
  39. T

    Abstract Algebra - Normal groups

    Homework Statement I'll be delighted to receive some guidance in the following questions: 1. Let G1,G2 be simple groups. Prove that every normal non-trivial subgroup of G= G1 x G2 is isomorphic to G1 or to G2... 2. Prove that every group of order p^2 * q where p,q are primes is...
  40. L

    Abstract algebra: Rings and Ideals

    Homework Statement The problem is to show that a subset A of a ring S is an ideal where A has certain properties. S is a ring described as a cartisian product of two other rings (i.e., S=(RxZ,+,*)). I have already proved that A is a subring of S and proved one direction of the definition of an...
  41. O

    Abstract Algebra: Is S(3) Isomorphic to Z(6)?

    Homework Statement Is the symmetric group s(3) isomorphic to Z(6), the group of integers modulo six with addition (mod 6) as its binary operationHomework Equations Basically i know that the symmetric group is all the different permutations of this set and that there are six of them. I also...
  42. K

    Solving a Question on Abstract Algebra - Help Needed

    need a help can't seem to figure out this one let 6 = (2,R) a. Find c [1 1] [1 0 ] b. c [o 1] [1 0] c find 2(6)
  43. D

    Exploring the Intuition Behind Rings in Abstract Algebra

    Ok so I am not a math major and i haven't taken an abstract algebra class but i am curoius about the subject. I have been watching video lectures at UCCS at http://cmes.uccs.edu/Fall2007/Math414/archive.php?type=valid and the proffessor talks about groups and rings. In the introduction the...
  44. W

    Abstract Alg. - maximal ideals in Z[x]

    Homework Statement Can someone tell me how can I prove that every ideal in Z[x] generated by (p,f(x)) where f(x) is a polynomial that is irreducible in Zp is maximal?? Thanks! Homework Equations The Attempt at a Solution
  45. W

    Abstract- Polynomial's Ring F[x]

    Homework Statement In this question we have to make use of the chinese remainder theorem and its applications: 1. Let F be a field and let p1(x), p2(x) two irreducible poynomials such as gcd(p1,p2)=1 over F. Prove that: F[x]/[p1(x)p2(x)] Isomorphic to F1 x F2 where F1=F[x]/(p1(x)) and...
  46. W

    Abstract Algebra Concept-based Question

    I have no abstract algebra background (only matrices and calculus and stats) but this problem came up in one of my classes and this time I'm completely clueless: Homework Statement A group is cyclic if an element, g, of the group generates the entire group in the sense that if h is any...
  47. T

    Abstract Alg.-Abelian groups presentation

    Homework Statement Let Cn be a cyclic group of order n. A. How many sub-groups of order 4 there are in C2xC4... explain. B. How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime? explain. C. Prove that if H is cyclic of order 8 then Aut(H) is a non-cyclic group. WHAT is...
  48. U

    Discussion group for abstract algebra? I'd be interested

    Hi everyone, I've just finished year 11 here in Australia and I've been reading some notes on abstract algebra just out of curiosity. I have had a little difficulty grasping the concepts, and I've read up on some linear algebra (up to the point of Euclidean n-space - haven't yet read about...
  49. T

    Isomorphism of Quotient Groups

    Homework Statement Let A,B be normal sub-groups of a group G. G=AB. Prove that: G/AnB is isomorphic to G/A*G/B Have no idea how to start...Maybe the second isom. theorem can help us... TNX! Homework Equations The Attempt at a Solution
  50. P

    Abstract Algebra: Commutative Subgroup

    Homework Statement Let G be a group and let a, b be two fixed elements which commute with each other (ab = ba). Let H = {x in G | axb = bxa}. Prove that H is a subgroup of G. Homework Equations None The Attempt at a Solution I'm using the subgroup test. I know how to show...
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