Abstract algebra Definition and 459 Threads

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...

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...

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...

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

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...

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 (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...

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...

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...

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...

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...

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 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...

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...

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 ∈ 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...

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 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...

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...

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...

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...

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 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 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 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

Hi, I have heard a few times that it is beneficial to study abstract algebra if I want to study computer science at advanced level (i.e. upper class, grad school, etc.), but is this true? If so, why would it be so? Thanks

Hi all. I am currently a Junior attending SUNY-Stony Brook as a math major. This coming fall semester I have a very good opportunity before me: I can either take the undergrad abstract algebra course (textbook: Contemporary Abstract Algebra by Gallian), or I can take the graduate abstract...

I've taken 2 (undergrad) courses in abstract algebra and a reading course in Galois Theory, and I still don't understand the point of studying groups and rings. The courses have not been particularly difficult for me, but my motivation is extremely low. In Galois Theory obviously I saw an...

Abstract Algebra- VERY SIMPLE but I don't understand what my HW question is asking! Homework Statement Hi. I am having trouble simply understanding what the question is here: (6) let w = (1 2 3 4 5 6 7 8 9 10 11 12 13 14). For which integers i is w^i a 14-cycle? Here is a link...