Abstract algebra Definition and 459 Threads

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...

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...

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...

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...

Homework Statement I'm trying to prove that this is a group. I already established elsewhere that it is a binary operation, so now I am onto proving associativity. I've tried many examples and so I'm confident it is associative, but now I just have to prove that.The Attempt at a Solution...

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...

Homework Statement Given that gcd(n,m)=1, prove that \mathbb Z_{nm}^\times = \mathbb Z_n^\times \oplus \mathbb Z_m^\times. Homework Equations / The Attempt at a Solution I can prove both groups have the same amount of elements (using Euler's totient function), but I can't figure out...

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?

Homework Statement Consider \frac{\mathbb Z_2[X]}{X^2+1}, is this ring isomorphic to \mathbb Z_2 \oplus \mathbb Z_2, \mathbb Z_4 or \mathbb F_4 or to none of these? Homework Equations / The Attempt at a Solution - \mathbb F_4 No, because \mathbb Z_2[X] is a principle ideal domain...

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)

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...

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...

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?

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...

Homework Statement Let D = Z[sqrt(10)], and let P be the ideal (2,sqrt(10)) 10). Prove that P is a prime ideal of D. Homework Equations The Attempt at a Solution Not sure where to start. I think elements are of the for a+b*sqrt(10), a,b integers. Any hints as to what to do next?

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 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

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...

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...

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.

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...

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...

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...

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...

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...

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?

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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...