Abstract algebra Definition and 459 Threads

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...

Homework Statement Let A = \left[ \begin{array}{cccc} 1 & 1 \\ 0 & -1 \end{array} \right] Let B = \left[ \begin{array}{cccc} 1 & 2 \\ 0 & -1 \end{array} \right] Find the smallest subgroup G of GL(n,R) that contains A and B. Also, find the smallest subgroup H of G that contains the matrices...

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...

The question asks: 3) Let X be the set of 2-dimensional subspaces of F_{p}^{n}, where n >= 2. (a) Compute the order of X. (b) Compute the stabilizer S in GL_{n}(F_{p}) of the 2-dimensional subspace U = {(x1, x2, 0, . . . , 0) ε F_{p}^{n} | x1, x2 ε F_{p}}. (3) Compute the order of S. (4)...

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...

Homework Statement prove that <x^m> intersection <x^n> = <x^LCM(m,n)> Homework Equations The Attempt at a Solution ===> let b be in <x^n> intersection <x^m> then for some t,k,p in Z, b=x^(mt) = x^(nk) thus b=x^(LCM(m,n) * p i.e. b is in <x^LCM(m,n)> <=== let b be...

Homework Statement Let c=cos(2pi/5). It can be shown that (4c^2)+(2c)-1=0. Use this fact to prove that a 72degree angle is constructible. Homework Equations The Attempt at a Solution I can see that using the equation and what c equals that you get the statement 0=0 and I know...

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...

(This is my first post on PF btw - I posted on this another thread, but I'm not sure if I was supposed to) I was doing some practice problems for my exam next week and I could not figure this out. Homework Statement Suppose a is a group element such that |a^28| = 10 and |a^22| = 20...

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...

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...