Abstract Definition and 506 Threads

Homework Statement Let f:R→S be a homomorphism of rings. If J is an ideal in S and I={r∈R/f(r)∈J}, prove that I is an ideal in R that contains the kernal of f. Homework Equations The Attempt at a Solution I feel like I have the problem right, but would like to have someone look...

Homework Statement Let G be a group with identity e. Let a and b be elements of G with a≠e, b≠e, (a^5)=e, and (aba^-1)=b^2. If b≠e, find the order of b. Homework Equations Maybe the statement if |a|=n and (a^m)=e, then n|m. Other ways of writing (aba^-1)=b^2: ab=(b^2)a...

Homework Statement Apply the division algorithm for polynomials to find the quotient and remainder when (x^4)-(2x^3)+(x^2)-x+1 is divided by (2x^2)+x+1 in Z7. Homework Equations The Attempt at a Solution I worked the problem and got that the quotient was (4x^2)-3x-1 and the...

Homework Statement Prove that the group Q/Z under addition cannot be isomorphic to the additive group of a commutative ring with a unit element, where Q is the field of rationals and Z is the ring of integers. Homework Equations The tools available are introductory-level group theory and...

I hear a lot that group theory is important to condensed matter physics. Does it have any practical use? Like if I were to do industry work in materials, would I ever use it? Is it important enough to take a full course on abstract algebra?

Homework Statement Homework Equations The Attempt at a Solution

Homework Statement If a is the only element of order 2 in a group G, prove that a is an element of Z(G). [Z(G) is the notation used by the book for center of group G] Homework Equations Z(G)={a is an element of G: ag=ga for every g that is an element of G} The Attempt at a...

Pick a number n which is the product of 2 distinct primes 5 or more. Find the number of elements of each order in the groupd D(sub)n+Z(sub)9, completely explaining your work. Verify that these number add up to the order of the group. Ive used 7 and 11 as my primes. So now do I use these...

well the title itself seems to be a paradox, but, What are some applications of abstract algebra (like groups, fields, and rings)? Apparently this determines the symmetry of particles in physics but what are some real-life, money-making application of group theory? (Yes, I money is one of my...

Abstract Algebra Questions... I have two problems that I'm a little puzzled by, hopefully someone can shed some light. 1) Show that if H and K are subgroups of the group G, then H U K is closed under inverses. 2) Let G be a group, and let g ε G. Define the centralizer, Z(g) of g in G to...

I was taught that the columns of a matrix, T, representing a transformation represent the first vector space's basis set and the rows represent the basis set of the range vector space. i.e. T(v_k) = t_1,k*w_1 +... + t_(m,k)*w_m So v_k would be the k-th basis vector of the first space, V...

Homework Statement The problem seems too easy so I suspect that I am overlooking something important. A problem this easy would be completely out of character for my professor...

I'm trying to look for papers with funny abstracts (in particular, there is one I saw a while ago, I believe it was physics, that had an abstract that only said no which I am trying to find, but I would love to see other funny one as well.)

1. Problem: Suppose a is a group element such that |a^28| = 10 and |a^22| = 20. Determine |a|. I was doing some practice problems for my exam next week and I could not figure this out. (This is my first post on PF btw) 2. Homework Equations : Let a be element of order n in group and let k...

Abstract Algebra Proof: Groups... A few classmates and I need help with some proofs. Our test is in a few days, and we can't seem to figure out these proofs. Problem 1: Show that if G is a finite group, then every element of G is of finite order. Problem 2: Show that Q+ under...

I've read up a little bit about Abstract Algebra and it seems like a really interesting subject. A university near me will offer an intro class in it next semester. Trouble is, the university requires Calc III as a prerequisite for the course. I'm taking AP Calc right now at school, but it...

Title: Can apparent superluminal neutrino speeds be explained as a quantum weak measurement? Abstract: Probably not http://arxiv.org/abs/1110.2832

I've put the problem and my attempt all in one image. I would show the above using induction. But to show the order of the element in question, we need to know what power to raise the element to such that it equals the identity. I feel like the problem is missing information about the...

I'll post the problem and my attempt at solution all in one picture: In the red step, I'm using commutative multiplication. Am I allowed to do this? I'm not sure, because the subset of G might not be a subgroup, so I don't know if its necessarily abelian like G is. Or does the fact...

Homework Statement Show that if H is a subgroup of G and K is a subgroup of H, then K is a subgroup of G. Homework Equations The Attempt at a Solution Well I know that H is a subgroup of G if H is non empty, has multiplication, and his inverses. So I assume that K is a subgroup...

So my understanding is an abstract class is one with at least one abstract method ( ie cannot be executed). It can therefore not bei nstantiated (not entirely sure what instantiated means) An interface I am a bit more confused about. first of all i thought it was where unrelated objects...

p<q, r<s, and r<q. Which of the following statements must be true? I. p<s II. s<q III. r<p The correct answer could be either one statement, a combination of statements, or none of the statements. Came across this question while helping some high school students prepare for their SATs...

Homework Statement A is a subset of R and G is a set of permutations of A. Show that G is a subgroup of S_A (the group of all permutations of A). Write the table of G. Onto the actual problem: A is the set of all nonzero real numbers. G={e,f,g,h} where e is the identity element...

Homework Statement Let G be a finite group and let x and y be distinct elements of order 2 in G that generate G. Prove that G~=D_2n, where |xy|=n. I have no idea how to solve this or even where to begin. I tried setting up G=<x,y|x^2=y^2=1=(xy)^n> But couldn't get any farther, I am so...

Homework Statement If (a,c) = 1 and (b,c) = 1, prove that (ab,c) = 1. Note that (x,y) refers to the greatest common divisor between x and y. 2. The attempt at a solution There is a theorem that says since (a,c) = 1, there exist integers u and v such that au + cv = 1. Likewise, there...

Homework Statement Let T be a subset of S and consider the subset U(T)={f \in A(S) | f(t)\inT for every t\inT}. 1) If S has n elements and T has m elements, how many elements are there in U(T)? 2) Show that there is a mapping F:U(T) -> Sm such that F(fg)=F(f)F(g) for f,g\inU(T) and F is onto...

Hello, I was wondering if if has any sense of talking about angles on an arbitrary http://en.wikipedia.org/wiki/Metric_space" (where only a distance function with some properties is defined) At first sight it seems to not has any sense, only some metric spaces has angles, namely does that...

It's been some time that I've been studying abstract algebra and now I'm trying to solve baby Herstein's problems, the thing I have noticed is that many of the exercises are related to number theory in someway and solving them needs a previous knowledge or a background of elementary number...

Homework Statement if f \in Sn show that there is some positive integer k, depending on f, such that fk=i. (from baby Herstein). The Attempt at a Solution Suppose that S={x1,x2,...,xn}. Elements of Sn are bijections from S to S. to show that fk=i it's enough to show that fk(xm)=xm for every...

Prove: If x has a right inverse given by a and a left inverse given by b, then a = b.The Attempt at a Solution One thing that bothers me: how can we even talk about a left inverse or a right inverse without establishing that x is in an algebraic structure? I wrote this in my proof but I'm not...

Homework Statement Let r,s,t and v be integers with r>0. If st=r+v and gcd(s,t)=r, then gcd(v,t)=r Homework Equations Just stumped. I am not sure what to do next.The Attempt at a Solution There are 2 integers d and e such that S=dR and T=eR, and 2 integers a and b such that Sa+Tb=R. I know I...

Can you recommend me to a book in Abstract Algebra and pre-requistes ?

Homework Statement Let R be an integral domain and suppose that R[x] is a principal ideal domain. Show that R is a field. Homework Equations I don't know where to start, I'm not familiar with this material. I was browsing through an abstract algebra book and found this. Would like...

Currently I am reviewing basic algebra, trigonometry and I will also be starting calculus this fall semester... I enjoy reading about math and I wanted to know what abstract algebra is? Would this be to difficult to read seeing that I am only starting calculus? If so what other types of...

Are there any basic prerequisites before learning about these branches of mathematics?

Hello, I just took ordinary diff eq and I've had calc III and linear algebra, but I'm worried about taking Modern Algebra or Real Analysis next semester because I have no experience writing proofs. The linear algebra class was all computation on tests and homework (we did see some proofs on...

The .pdf can be ignored. Let A + B = (A - B) U (B - A) also known as the symmetric difference. 1. Look for the identity and let e be the identity element A + e = A (A - e) U (e - A) = A Now there are two cases: 1. (A - e) = A This equation can be interpreted as removing from A all elements...

I am undecided between these two for 2012 spring term (my last semester hopefully)

I first introduce the vector along the lines 'something with magnitude and direction'. Later on the definition of a vector becomes generic - 'an element of a vector space'. Euclidean spaces (n=2 and n=3) are something we can all visualize. However when describing other vector spaces such as...

Homework Statement Let G be a group with pk elements, where p is a prime number and k is greater than or equal to 1. Prove that G has a subgroup of order p. The Attempt at a Solution I attempted to prove this by showing that the conditions for a set to be subgroup form a subgroup of order p...

Use of "I" vs "we" in abstract I'm writing an abstract for a poster which I am required to give for a fellowship. In general it is my understanding that no one would use "I" in an abstract, even if he or she were the sole author. But in this case, the purpose of the poster session is to...

I really feel dissapointed in myself that I didn't perform as well as I wanted last semester. I took Modern Algebra I and Geometry. The Geometry class covered Euclidean and non-Euclidean geometries. I bombed the final but earned an overall of a B+ because of a 90-something percentile homework...

Hello everyone. I am graduating high school in about a few days (yay!). I'm going to enter university this fall, majoring in math. I just want a few advice on what to do this summer to prepare myself for college mathematics. How can I prepare myself to tackle abstract math, without any prior...

I've been studying cryptography and I found out that AES uses Galois Fields. I was therefore wondering where else does abstract algebra pop-up for real world use?

In today's world its not enough just to learn things. You have to be able to learn them fast or you will never accomplish very much. Mathematics can of course be quite difficult to wrap your head around at times and so if one can learn mathematics quickly and efficiently then there are few...

What is the absolute best abstract algebra book for graduate students? I was wanting a book that covers algebra in the most comprehensive manner possible, at about the level of Hungerford's Algebra. I was wondering if Carstensen's Abstract Algebra in the Sigma Series in Pure Mathematics is a...

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...

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?

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

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