Abstract Definition and 506 Threads

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...

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...

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 =...

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...

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...

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...

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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.

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...

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

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...

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...

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 ∈ S[SIZE="1"]17 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...

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...

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...

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

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...

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...

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...

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)

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...

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

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...

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...

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...

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...

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

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...

I just finished my first quarter of analysis (Text: Rudin's PMA) and abstract algebra (Text: Beachy and Blair) courses. I must say I really enjoyed these courses, and I feel like I learned a lot from them. However, I still ended up getting B+'s from both of these courses. While I'm not...

1. If G is a finite group that does not contain a subgroup isomorphic to Z_p X Z_p for any prime p. prove that G is cyclic im stumped. i don't understand the 'does not contain a subgroup isomorphoc to Z_p X Z_p part. ive tried using cauchy's theorem for abelian group: if G is a finite...

Homework Statement The problem is: Let G be a group of order 12 ( o(G)=12). Let's assume that G has a normal sub-group of order 3 and let a be her generator ( <a>=G ). In the previous parts of the questions I've proved that: 1. a has 2 different conjucates in G and o(N(a))=6 or...

Homework Statement Solve the inhomogeneous differential equation dX/dt=AX+B in terms of the solutions to the homogeneous equation dX/dt=AX. Homework Equations A is an nxn real or complex matrix and X(t) is an n-dimensional vector-valued function. If v is an eigenvector for A with...

Homework Statement Let F be a field of characteristic p>0 and let E = F(a) where a is separable over F. Prove that E=F(a^p). Homework Equations The Attempt at a Solution I know that maybe show how mod F(a) = mod F(a^p) or something around there.

Question: Prove the following properties of cosets. Given: Let H be a subgroup and let a and b be elements of G. H\leq\ G Statement: aH=bH \ if\ and\ only\ if\ a^{-1}b\ \epsilon\ H The statement is what I have to prove. My issue is I don't know how to start off the problem. When I...

For this problem, I have to find all orbits of given permutation. \sigma: \mathbb{Z} \rightarrow \mathbb{Z} Where, \sigma(n)=n-3 Now, the problem is I do not know how to approach this permutation in the given format. All the permutations I dealt with were in the form: \mu...

Homework Statement List all the elements of GL_N(\mathbb{Z}/2\mathbb{Z}). Find the order of each element, and show it is not abelian. The Attempt at a Solution I am confused right from the get go about GL_n(\mathbb{Z}/2\mathbb{Z}). I think the L_n(\mathbb{Z}/2\mathbb{Z}) part...

Hey all, I`ve been working at this "proof" for several hours now, have put it away several times thinking that maybe I`ll get it if I leave it alone for a bit...has not worked =] It has 2 parts, I think I have proven the first part, but the second one really just stumps me =| 1. Show that a...

Homework Statement prove that if 2 does not divide a then 24 divides a^2-1 Homework Equations I know that if 2 does not divide a then a is odd. I proved that the square for all odd integers are of the form 8K+1 I also proved the square of any integer is either in the form...

How do you know if one is good at abstract thinking? I'm self evaluating myself if I can do pure maths in college, so any advice would be appreciated.

How hard is Artin's Algebra book to understand? For a student who has not had any upper level (proof based) math classes beyond calculus, is it doable if you are sufficiently motivated?

Hi everybody, I'm new to absract algebra and I really can not understand different between direct sum and direct product in group theory (specially abelian groups). could does anyone give me a clear example or ... ? thanks

Homework Statement Let {S1,L1} and {S2,L2} be abstract geomettries. If S=S1 ^ S2 and L=L1 ^ L2 prove that {S,L} is an abstract geometry ( where ^ = intersection) Homework Equations The Attempt at a Solution Let {S1,L1} and {S2,L2} be abstract geometries. Assume that S=S1 ^...