Abstract algebra Definition and 459 Threads

Homework Statement (a) Suppose a belongs to a group and lal=5. Prove that C(a)=C(a3). (b) Find an element a from some group such that lal=6 and C(a)≠C(a3). Homework Equations The Attempt at a Solution For (a) I know I need to show that every element in the set C(a) is...

Homework Statement For any integer n>2, show that there are at least two elements in U(n) that satisfy x2=1. Homework Equations In my book I found a definition: Define U(n) to be the set of all positive integers less than n and relatively prime to n. The Attempt at a Solution...

Homework Statement This problem is from Charles C. Pinter's A Book of Abstract Algebra, Second Edition. The problem is B7 of Chapter 2.Show that the operation * is either associative or not. x*y=\frac{xy}{x+y+1} This problem seems simple to me: I keep arriving at YES for an answer...

Hello , I've a question . How Can someone Get Harvard Books on Mathematics generally and Abstract Algebra Specially ? How Can The one Buy it ?

Hey guys, As of now, I am in a sets and logic proof based course (Intro to proof-writing). This course basically teaches logic, how to write proofs using examples of algebraic equations, sets: power sets, unions and intersections of classes, etc. With a C in this course, you can register...

Hello all, In the Fall I am planning on taking Real Analysis, Abstract Algebra and doing an independent study in something(my professor has yet to get back to me on what he is willing to do it in). My question is would it be too much of a workload to try and do another independent study in...

currently i am a math major, still unsure whether pure or applied. i am also looking to double major in physics. which class would be more helpful to me: abstract algebra, or (upper division) ODE class? I have taken the lower division DE class already.

Hello all, I am currently doing a self-study in Abstract Algebra. I was a math major in college (not so long ago), so I have some exposure to upper level math. For one reason or another, I wanted to go back and re-learn Abstract. I was using Fraleigh until I discovered Pinter's text which...

I need to buy a textbook for self study in abstract algebra for self study. Although I'm a physics major, I have lot's of experience with proof. I'm between Artin's Algebra and Dummit's Abstract Algebra. Which one do you recommend?

How difficult is abstract algebra or group theory, plus complex analysis in relation to calculus?

Homework Statement M = {(pa,b) | a, b are integers and p is prime} Prove that M is a maximal ideal in Z x Z Homework Equations The Attempt at a Solution I know that there are two ways to prove an ideal is maximal: You can show that, in the ring R, whenever J is an ideal such...

https://www.amazon.com/dp/1111569622/?tag=pfamazon01-20 Any idea?

Homework Statement f is a quadratic function from the second degree and f(a)=bc;f(b)=ac;f(c)=ab Homework Equations Calculate : f(a+b+c) The Attempt at a Solution Can we say that f(a+b+c)=f(a)+f(b)+f(c) and the go on from there plugging in the values of each one are do i have to do...

Help Develop "Tongue in Cheek" Abstract Algebra Proof Hello all, First and foremost I would like to thank everyone on the forum. Your post here have been invaluable in aiding me in completing many of my engineering courses. Now I am attempting to develop a "tongue in cheek" proof using...

Homework Statement Let K \subseteq L be fields. Let f, g \in K[x] and h a gcd of f and g in L[x]. To show: if h is monic then h \in K[x]. The Attempt at a Solution Assume h is monic. Know that: h = xf + yg for some x, y \in K[x]. So the ideal generated by h, (h) in L[x] equals...

Homework Statement Let G be an abelian group of order n. Define phi: G --> G by phi(a) = a^m, where a is in G. Prove that if gcd(m,n) = 1 then phi is an isomorphism Homework Equations phi(a) = a^m, where a is in G gcd(m,n) = 1 The Attempt at a Solution I know since G is an...

hey, I have this group I've been trying to generate using the GenerateGroupoidByRelations[] function but it keeps giving me an error, G = GenerateGroupoidByRelations[{a, b}, {a^4 == e, b^4 == e, a ** b ** a ** b == e, a^3 ** b ** a^3 ** b == e}, SizeLimit -> 60] gives...

Actually I'm stupid today, it happens once in a while that I get extremely lazy and stupid in mathematics, but today I came up with a bizarre thing in abstract algebra that I couldn't find my mistake on my own and I'm not sure whether what I've concluded is true or wrong, I was proving another...

This is not really a homework questions, rather a concept based one. I am studying from Fraleigh's ''Intro to abstract algebra'' and in chapter 15 it states, that for a group G and normal non-trivial subgroup of N of G, the factor group G/N will be smaller than G. I am not sure how he counts the...

http://i111.photobucket.com/albums/n149/camarolt4z28/untitled.jpg G/N is the set of all left cosets of N in G. I don't understand the notation. a) The permutations are (1,2), (2,3), (3,1). What are the left cosets - <1>, <2>, <3>? That doesn't make sense with permutations. b) I have...

Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.

Homework Statement Let R be a ring and a,b be elements of R. Let m and n be positive integers. Under what conditions is it true that (ab)^n = (a^n)(b^n)? Homework Equations The Attempt at a SolutionWe must show ab = ba. Suppose n = 2. Then (ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = (aa)(bb) =...

Homework Statement The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so...

The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so I am pretty lost in...

Homework Statement Show that the center of a Clifford algebra of order 2^n is of order 1 if n is even, and 2 if n is odd Homework Equations the center of an algebra is the subalgebra that commutes with all elements Clifford algebra of 2^n is defined as being spanned by the bases...

Let G be a non-cyclic group of order pn where p is a prime number. Prove that G has at least p+3 subgroups. Could anyone offer a solution to this problem?

Hi. I've just failed my first test in my Abstract Algebra course... I'm sure I scored a zero. So... needless to say, I need help. Do you know of any good websites with lots of examples? Or even a really good book with lots of problems? The textbook we're using is 'A First Course in Abstract...

Show that \langle a,b \rangle = \langle a,ab \rangle = \langle a^-1,b^-1 \rangle for all a and b in a group GI am not sure what this question is asking. Does this notation mean that a the cyclic group is generated by a,b and any combination of the two?

1. There should be a separate (sub)forum for NT. ... and one for abstract algebra, for that matter! 2. Show that there are infinitely many n such that both 6n + 1 and 6n - 1 are composite. Without CRT, if possible. My work... let n = 6^{2k}. Then 6n \pm 1 = 6^{2k + 1} \pm 1... Hmm. Having a...

Homework Statement a.) Let a=3-8i and b=2+3i. Find x,y ϵ Z[i] such that ax+by=1. b.) Show explicitly that the ideal I=(85,1+13i) \subseteq Z[i] is principle by exhibiting a generator. Homework Equations Given ideal: I=(85,1+13i) \subseteq Z[i] a=3-8i b=2+3i Honestly, I am beyond lost...

Homework Statement Let a=p_{1}^{r_{1}}p_{2}^{r_{2}}...p_{k}^{r_{k}}, b=p_{1}^{s_{1}}p_{2}^{s_{2}}...p_{k}^{s_{k}} where p_{1},p_{2},...,p_{k} are distinct positive primes and each r_{i},s_{i} ≥ 0 Prove that (a,b)=p_{1}^{n_{1}}p_{2}^{n_{2}}...p_{k}^{n_{k}} \mbox{ where for each } i...

Homework Statement How do I determine the parity of a permutation? I think my reasoning may be faulty. By a theorem, an n-cycle is the product of (n-1) transpositions. For example, a 5 cycle can be written as 4 transpositions. Now say I have a permutation written in cycle notation: (1...

I'm an undergrad math major, and this is my first semester taking upper level math. I'm currently taking Abstract Algebra, and feeling pretty intimidated. I mean, I feel out of the loop, I'm trying hard to understand, but I feel overwhelmed, like maybe it's too much for me. Is it normal to feel...

Hi, I'm doing a Physics undergrad and this semester I have the following courses: Thermodynamics, Quantum Mechanics, Numerical Methods, an Astrophysics course, and a Computational Lab. I've also taken Abstract Algebra which has twice as many lectures as any of these. Add to this the fact that I...

Write the following in two row matrix form. (1874)(36759) I have [1 2 3 4 5 6 7 8 9] [8 2 6 1 9 7 4 7 3] my problem is couldn't 7 also go to 5 and have 8 going to 7 and 6 going to 7 so I am sure I am wrong but I am not sure why.

Does anyone here know any good websites or sources to help me learn and understand abstract ablegra?

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...

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...

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...

Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...

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...