Abstract algebra Definition and 459 Threads

Homework Statement Let R be an Integral Domain. Prove that if a,b are elements of R and both a and b are units in R, then prove a*b is a unit of R. Homework Equations a is a unit in R if and only if there exists an element u in R such that au=1=ua where 1 is the identity element of R...

Homework Statement If G1, G2 are two groups and G = G1 times G2 = {(a,b) such that a is an element of G1, b is and element of G2}, where we define (a,b)(c,d) = (ac, bd), (a) Show that N = {(a, e2) such that a is an element of G1}, where e2 is the unit element of G2, is a normal subgroup...

Homework Statement If f is a homomorphism of G onto G' and N is a normal subgroup of G, show that f(N) is a normal subgroup of G'. Homework Equations The Attempt at a Solution Once again, I'm completely lost.

Homework Statement Let F be a field and f(x) in F[x]. If c in F and f(x+c) is irreducible, prove f(x) is irreducible in F[x]. (Hint: prove the contrapositive) Homework Equations So, I am going to prove if f(x) is reducible then f(x+c) is reducible. The Attempt at a Solution f(x)...

Abstract Algebra -- lifting up a factor group After spending an extended period with my Professor during office hours I must admit I am mystified. He kept on talking about "lifting up" factor groups. I think this has something to do with using a factor group, say G/N, to show that there...

Abstract Algebra -- no Sylow allowed Please note Sylow's theorem(s) may not be used. Using Theorem 1 as a tool, prove that if o(G)=p^{n}, p a prime number, then G has a subgroup of order p^m for all 0\leq m\leq n. Theorem 1: If o(G)=p^{n}, p a prime number, then Z(G)\neq (e). Theorem 1 uses...

Abstract Algebra -- group Show that in a group G of order p^2 any normal subgroup of order p must lie in the center of G. I am pretty sure here that p is supposed to be a prime number, as that is the stipulation in preceding and later problems. However, the problem statement does not...

I have two problems I would like to discuss. 1.For any group G prove that the set of inner automorphisms J(G) is a normal subgroup of the set of automorphisms A(G). Let A be an automorphism of G. Let T_{g} be an inner automorphism, i.e. xT_{g}=g^{-1}xg The problem can be reduced to the...

1) find a group that contains elements a and b such that ︱a︱=︱b︱= 2 and a) ︱ab︱ = 3 b) ︱ab︱=4 c) ︱ab︱=5 2) suppose that H is a proper subgroup of Z under addition and H contains 18, 30 and 40. determine H? does anyone can help me out? and ...i am really in...

Abstract Algebra -- isomorphism question If N, M are normal subgroups of G, prove that NM/M is isomorphic to N/N intersect M. That's how the problem reads, although I am not sure how to make the proper upside-down cup intersection symbol appear on this forum. Or how to make the curly "="...

Homework Statement I am asked to show that if E is a semi-group and if (i) there is a left identity in E (ii) there is a left inverse to every element of E then, E is a group.The Attempt at a Solution Well I can't seem to find the solution, but it's very easy if one of the two "left" above is...

Homework Statement Prove that if (ab)^2=a^2*b^2, in a group G, then ab =baHomework Equations No equations necessary for this proofThe Attempt at a Solution Suppose (ab)^2=a^2*b^2. Then (ab)^2=(ab)(ab)=(abba)=(ab^2*a)=a^2 *b^2=> (ab)(ba)=(ba)(ab) = e By cancellation, (ab)=(ba) <=> (ba)=(ab)

Homework Statement Show that {1,2,3} under multiplication modulo 4 is not a group but that {1,2,3,4} under multiplication modulo 5 is a group Homework Equations a mod n=r ;a=qn + r The Attempt at a Solution I'm going to assume when the problem says modulo 4, the problem is read...

I'm thinking about taking the math GRE in December but I've never studied abstract algebra--all this about rings and groups just flies right over my head. Can anyone recommend a good introductory book? I'm thinking one of the Dover works might be good since they seem to emphasize problem...

I am currently signed up for an intro abstract algebra course. I will be taking this course and calculus 3(stewart's book). I am pretty good at writing proofs. Do you have to know calculus 3 to do well in abstract algebra? Or can you take it concurrently? Is abstract algebra considered a...

This question links to a former discussion on the board. I'm confused regarding this thread: https://www.physicsforums.com/showthread.php?t=3622" Specifically, towards the end of the thread, the asker states (in regards to the union notation originally cited): "...if we say that x is an...

Another Abstract Algebra Question... Every symmetry of the cube induces a permutation of the four diagonals connecting the opposite vertices of the cube. This yields a group homomorphism φ from the group G of symmetries of the Cube to S4 (4 is a subscript). Does φ map G onto S4? Is φ 1-1? If...

Abstract Algebra Questions... Help Please! Any and all help on these problems would be greatly appreciated. Thank you in advance to any who offer help :smile:. 1. Let φ:G->H be a group homomorphism, where G has order p, a prime number. show that φ is either one-to-one or maps every element...

G is a finite group, |G| =p^n, p prime *:GxX -> X is group action. X is a finite set, I am required to prove the following |X|\equiv |X^G|modp Now we start by asserting that x_1, x_2, ...,x_m is the set of m orbit representatives. That orbit x <x_i> = {x_i} \\ iff x_i is a...

Homework Statement Prove that (a b c d) is a unit in the ring M(R) if and only if ad-bc !=0. In this case, verify that its inverse is (d/t -b/t -c/t a/t) where t= ad-bc. Homework Equations An element a in a ring R with identity is called...

Homework Statement From An Introduction to Abstract Algebra by T. Hungerford Section 3.2 #29 Let R be a ring with identity and no zero divisors. If ab is a unit in R prove that a and b are units. Homework Equations c is a unit in R if and only if there exists an...

Please I need your help for that qustion and how do slove that qustion's problem. can you help me for slove for that? Pleasee Let G be any group, and let a, b ∈ G. Show that there are c, d ∈ G such that ac = b and da = b. Hint: you have to give an explicit definition for c and d in terms of a...

ProbelmLet p and q be distinct primes. Suppose that H is a porper subset of the integers and H is a group under addition that contains exactly three elements of the set {p, p+q, pq, p^q, q^p}. Determine which of the following are the three elements in H. a.pq, p^q, q^p b. p+q, pq, p^q c. p...

I'm having trouble understanding splitting fields. Some of the problems are find the degree of the splitting field of x^4 + 1 over the rational numbers and if p is a prime prove that the splitting field over the rationals of the polynomial x^p - 1 is of degree p-1. I'm really confused with these...

can anyone help me with my abstract algebra assignment? Let a be an fixed element of some multiplicative group G. Define the map β: Z > G from the additive inter group Z to G by β(n)=a^n. i. Prove that the map β is a homomorphism. ii. Prove/Disprove that the map β is an isomorphism. thanks!

In my uni I am forced to make a painful choice btw taking PDE or abstract algebra. I will take algebra, but I'd like to know what I will be missing? What is being taught in this class exactly? (BESIDES HOW TO SOLVE A PDE BY SEPARATION OF VARIABLES :rolleyes:)

I'm supposed to find a non-trivial group G such that G is isomorphic to G x G. I know G must be infinite, since if G had order n, then G x G would have order n^2. So, after some thought, I came up with the following. Z is isomorphic to Z x Z. My reasoning is similar to the oft-seen proof...

Can someones tells me how to prove these theorems. 1. Prove that if G is a group of order p^2 (p is a prime) and G is not cyclic, then a^p = e (identity element) for each a E(belongs to) G. 2. Prove that if H is a subgroup of G, [G:H]=2, a, b E G, a not E H and b not E H, then ab E H. 3...

Given R=all non-zero real numbers. I have a mapping Q: R-> R defined by Q(a) = a^4 for a in R. I have to show that Q is a homomorphism from (R, .) to itself and then find kernel of Q. In order to prove homomorphism i did this, for all a, b in R Q(ab) = (ab)^4 = a^4b^4 = Q(a)Q(b). Is...

1) prove that H is a subgroup of S5 (the permutation group of 5 elements). every element x in H is of the form x(1)=1 and x(3)=3, meaning x moves 1 to 1 and moves 3 to 3. does your argument work hen 5 is replaced by a number greater than or equal to 3? 2) Let G be a group. prove or disprove...

Hi. My latest question concerns the following. I must prove that the alternating group A_n contains a subgroup that is isomorphic to the symmetric group S_{n-2} for n = 3, 4, ... So far, here's what I have (not much). The cases for n = 3 and n = 4 are elementary, since the group lattices...

Abstract algebra help please! I'm not sure if I've posted in the correct forum but I would like some help with the following question: http://i12.tinypic.com/33mlik6.jpg" I've to complete this table but I am unsure of how to do the very first step which is to fnd wv. I am learning this...

Hi I'm fairly new at abstract algebra and have therefore got stuck with this assignment. Hope there is somebody here who can help me complete it, because I have been ill these last couple of weeks. Its goes something like this b is a number written in base 10 b\;= \;b_010^0 +...

I have finally understood algebra. It is all about understanding groups. first of all we tackle abelian groups, and finitely generated ones at that., we completely classify them as sums of cyclic groups using their structure as Z modules where Z is the integers. then we ask about non...

This is question 53\gamma. Given a group G of transformations that acts on X... and a subgroup of G, Go (g * x = x for all x for each g in Go), show that the quotient group G/Go acts effectively on X. A group G "acts effectively" on X, if g * x = x for all x implies that g = e, where g is a...

hello i have two questions and i need answers for them first one: in the additive group (Z,+) show that nZ intersection mZ= lZ , where l is the least common multiple of m and n. The second question is : Given H and K two subgroups of a group G , show the following...

Lately I've been taking a unit that deals with abstract algebra and I'm finding myself not understanding the lectures at all. To make matters worse the unit doesn't have a reccomended textbook so I don't even have any infomation to self learn from. I guess what I'm asking is for some good...

Hi everyone; There are some questions which are frizzling my mind, if anybody could help then please reply to these ques which are as follows. Q1) Prove that homomorphic image of cyclic group is itself cyclic? Q2) Prove that any group 'G' can be embedded in a group of bijective mapping of...

i reackon youv'e seen it already, the problem is to rearrange the next numbers in the fixed order: 1 2 3 4 5 6 7 8 9 11 10 when you have at the last entry a vacant place you need to put it in order. this is from the text of edwin h. connell, and i think it's impossible (after a lot...

Hello, everyone, i am a newbie here. I am currently taking a modern linear algebra course that also focus on vector spaces over the fields of Z[SIZE="1"]p and complex numbers. Since i am not familiar with typing up mathematics using tex or anything so that i can post on the forums, i will use...

I just started with the course of discrete mathematics,,where we have abstract algebra..I am actually interested in the application of this algebra..i know that this is used in Cryptography,error correcting codes,and theoritical computer science...i just want to have a basic outline of how they...

Does anyone know a good Abstract Algebra text? I currently have the text by Gallian and quite frankly I give two stars out of five. I'm looking for something that's more advanced, and well written. Any recommendations will be greatly appreciated. :biggrin:

Hello. We got a review today in abstract algebra, and I am stuck on two problems. 1) Let f: G -> H be a surjective homomorphism of groups. Prove that if K is a normal subgroup of G, then f(K) is a normal subgroup of H. Where f(K)= {f(k): k \inK} The entire f(K) part is really throwing me...

I'm 17 and my high school has no other math courses to offer me. At a local college there is a course called "Modern Algebra" and I was wondering if it was the same thing as abstract algebra. I asked my math teacher about it, and he said it was the hardest math class he took; he used to call...

Help with abstract algebra! Here is my quetion. What is the order of the set of all 2x2 matricies (such that its entries a,b,c,d are between 0 and p-1), and whose determinant is congruent to 1 modulo p ? => Order of SL(2,Fp) thanks :)

Hello. I was reading a journal and an interesting problem came up. I believe the journal was in the American Mathematics Society publications. Well, here's the statement. "For all integers, n greater than or equal to 3, the number of compositions of n into relatively prime parts is a...

define right-inverse of a mapping B to be mapping A, such that B * A= identity (iota). Where the operation * is composition. Note that B is A's left-inverse. QUESTION: Assume S is a nonempty set and that A is an element of M(S) -the set of all mappings S->S. a) Prove A has a left...

This coming fall semester, I have a choice between taking Analysis and Abstract Algebra. Unfortunatly, I'm having a great deal of trouble deciding which to take. Both seem interesting (though Algebra more so). On the other hand, Analysis would open more options for spring semester (in...

I need some recommendations for a good linear algebra textbook, something that's actually used in schools. I've finished linear 1 and 2 and I'm doing some preparation during the summer.

I was given a problem to prove there are at most 3 groups of order 21, with extra credit for proving there are at most 2. I am pretty stuck on this one but here is what I have so far: Suppose G is a group of order 21 Let K be a sylow 3-subgroup of G and let H be a sylow 7-subgroup of G...