Abstract algebra Definition and 459 Threads

Homework Statement Coproducts exist in Grp. This starts on page 71. of his Algebra. Homework Equations [/B] Allow me to present the proof in it's entirety, modified only where it's convenient or necessary for TeXing it. I've underlined areas where I have issues and bold bracketed off my...

Dear Physics Forum advisers, Today, I got two gifts from my research mentor: "Topics in Algebra" by I.N. Herstein and "Abstract Algebra" by Dummit/Foote. I am very happy and grateful for his gifts, but I already have been studying the abstract algebra through Michael Artin and Hoffman/Kunze...

A couple of notes first: 1. \hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this. 2. This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes' 3. Titles are short, so I know that the hom-functor...

Dear Physics Forum advisers, I am a rising college junior in U.S. with a major in mathematics, and an aspiring applied mathematician. I apologize for this sudden interruption, but I wrote this email to seek your advice on my current problem on the course selection. I will very soon be...

This isn't homework, I'm just trying to refresh my memory on cyclic groups. My question is, in this problem solution, how does ##{\sigma_i}^m=1## follow from ##\sigma_i## being disjoint?

Homework Statement Hello guys So I have the following problem, given the mapping above I have to check weather it's ring homomorphism, and maybe monomorphism or epimorphism. The Attempt at a Solution So the mapping is obviously well defined, and I have proven it's homomorphism, and it's...

Dear Physics Forum advisers, My name is Phoenix, a sophomore with major in mathematics and an aspiring applied mathematician in the theoretical computing. I wrote this email to seek your recommendation on the textbooks for abstract algebra. I want to self-study the abstract algebra during...

Homework Statement Let F be a finite field of characteristic p∈{2,3,5}. Consider the quaternionic ring, Q_F={a_1+a_i i+a_j j+a_k k|a_1,a_i,a_j,a_k ∈ F}. Prove that Q_F is not a division ring. Homework EquationsThe Attempt at a Solution Let α=1+i,β=1+i+j∈QF. Then...

Homework Statement Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer. Homework EquationsThe Attempt at a Solution Let ##\alpha= \sqrt{2}+\sqrt{3}\in \mathbb{R}##, then ##\alpha^4-10\alpha^2+1=0## which is a root of ##f(x)=x^4-10x^2+1## where ##f(x)## in...

Homework Statement Construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. How many non-zero elements are primitive in this field? Calculate $|GL2_(\mathbb{F}_16)|$. Homework Equations Primitive Theorem The Attempt at a Solution For the first question, I don't know how to construct...

Dear Physics Forum advisers, I am a college sophomore in US with a major in mathematics, and an aspiring algebraic number theorist and cryptographer. I wrote this email to seek your advice about taking the Analysis I (Real Analysis I), Abstract Algebra I, and Linear Algebra with Proofs. At...

Homework Statement Let F be a field. Show that F is isomorphic to F/{0} Homework EquationsThe Attempt at a Solution By the first ring isomorphic theorem, kernel of the homomorphism is an ideal which is either {0} or I. Hence F isomorphic to F/{0} I think I misunderstood the problem can...

Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework EquationsThe Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this...

Hi, Are Calculus I, II, III courses a prerequisite requirement for studying Abstract Algebra? I have read that Proofs and a willingness to work hard is. I am studying Logic and Set Theory and want to study Abstract Algebra in the distant future. I am focused on Foundational and Pure...

Homework Statement Let G be any group and a in G, define f: Z → G by f(n) = a^n Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z Homework EquationsThe Attempt at a Solution The range of f is a^n , then quotient group of Z is Z/nZ Apply the first...

I am planning to study the following pure mathematics areas (on my own) and wanted to know if this is the best sequence: 1- Formal Logic 2 -Philosophical Logic 3- Sentential Logic 4- Predicate Logic 5- Symbolic Logic 6 -Set Theory 7 -Pure Mathematics (Intro, Pure Math I and II and Hardy) -...

Hello, A couple of years ago I studied abstract algebra from Dummit and Foote. However, I was never able to gain the intuition on the subject that I would like from that book. I want to study the subject again, and I want to use a different book this time around - one that covers a lot of...

Hi I recently read a book called "The fundamental theorem of algebra" by Fine and Rosenberger. It focused specifically on polynomials, and proved the theorem using several fields of mathematics; Two of the proofs were algebraic. Abstract algebra has been very difficult for me; Mostly because...

Homework Statement Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+) Homework Equations Isomorphism, Group Theory, Binary Operation The Attempt at a Solution Let a,b,c,d∈ℝ Define f : M→ℂ by f( (a -b) (b a) ) = a+bi 1-1: Suppose f( (a -b) (b a) )= f( (c...

I found out I can pick up a second major in math should I elect to take a two semester sequence in abstract algebra. My first major is in chemical engineering. Right now, I plan on taking a two semester sequence in either: 1) probability with measure theory, 2) abstract algebra (Dummit and...

Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?

Homework Statement Let α = (α1α2...αs) be a cycle, for positive integers α1α2...αs. Let π be any permutation that παπ-1 is the cycle (π(α1)πα2...π(αs)). Homework EquationsThe Attempt at a Solution I started by choosing a specific α and π, and tried finding παπ-1 to give myself some idea of...

Homework Statement Find a normal subgroup H of Zmn of order m where m and n are positive integers. Show that H is isomorphic to Zm. Homework EquationsThe Attempt at a Solution I am honestly not even sure where to start. My initial thoughts were if Zmn was isomorphic to Zm x Zn then I could...

I'm a graduate mathematics student and I did my undergrad in applied math. I also took the normal 10 hrs of physics foundations and then a semester of modern physics (basic quantum intro, special relativity, orbit states etc.). I was thinking about pursuing study in areas that would be...

I am working on this problem with lots and lots of nesting definitions like this following, and I have been trying to get help from here as well as http://www.quora.com/How-do-I-prove-G-Z-R-G-is-isomorphic-to-Aut-R-G , but none gave me complete help: Show that ##G/Z(R(G))## is isomorphic to a...

I have this problem on simple group's homomorphism: Let ##G′## be a group and let ##\phi## be a homomorphism from ##G## to ##G′##. Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G′## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subset N##. And last year somebody...

Entering my third year of my bachelor of science majoring in maths/physics and having some trouble deciding what courses to do this semester. I know for sure I will be taking complex analysis and 3rd year quantum however am having trouble picking between 3 in particular for my final two courses...

To make a very long story short, in a group theory problem I am working on, I need to prove this: ##A \lhd B \Rightarrow A'\neq A##, where ##A## and ##B## are finite and ##A'## is called the commutator subgroup: ##\begin{align} A' :&= [A, A] \\ &= \langle [x, y] \mid x, y \in A \rangle \\ &=...

Should I take abstract algebra. I was going to double major but I don't want to be at school for more than four years or pay for extra classes. Therefore, I decided minor in mathematics instead. I registered for abstract algebra before I decided to just minor in mathematics. I have a hard time...

I am working on a problem on automorphism group of radical of finite group like this one: Here are what I know and what I don't know: ##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the...

I am self-studying a class note on finite group and come across a problem like this: PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).## Where ##O_p(G)## is the subgroup generated by all subnormal p-subgroups of ##G##; ##E(G)## is the...

I am working on myself on a problem looks like this: Let ##G'## be a group and let ##\phi## be a homomorphism from ##G## to ##G'.## Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G'## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subseteq N##. I have been asking...

I came across this problem in class note but I was stuck: Assume that ##G## be a group of order 21, assume also that ##G'## is a group of order 35, and let ##\phi## be a homomorphism from ##G## to ##G.'## Assume that ##G## does not have a normal subgroup of order 3. Show that ##\phi (g) = 1##...

Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...

Homework Statement Let V be a finite-dimensional real vector space with inner product <⋅,⋅> and L: V → R a linear transformation. Show that there exists a unique vector a ∈ V such that L(x) = <a,x>. Homework Equations Hey everyone, so I'm a physics student who had to choose a few electives in...

Not quite sure, could someone kindly explain the concept of the topic to me? Thanks, it really means a lot. :)

Having just finished an introductory course on group theory (with some bits of ring and field theory), I am completely enthralled with this type of math. I initially planned on taking Complex Analysis next semester since so many people say it's "useful" for physics (this was also a compromise...

< Mentor Note -- thread moved to HH from the technical math forums >[/color] My final assignment in graduate algebra is to write an essay about the relationship among the subjects we have learned so far this semester: (1) Module (2) The Field of Fractions of an Integral Domain (3) Integrality...

Hello, I'm debating between taking either abstract algebra, theory of numbers, or intermediate symbolic logic as a math elective. Does anyone have any idea which would make my life easier?

I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for pointers. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, thank you for...

I am reading a graduate-level Abstract Algebra lemma on noetherian integral domain, I am bring it up here hoping for help. The original passage is in one big-fat paragraph but I broke it down here for your easy reading. Let me know if I forget to include any underlying lemmas, and especially...

Homework Statement Let ab=a and ba=b, show that a^2 = a and that b^2 = b Homework Equations none The Attempt at a Solution Not sure if I did this correct.. but here is what I did. Given: ab = a. Multiply both by left hand multiplication by a^-1 a^-1*a*b = 1. where a^-1*a is obviously...

Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.

Hi, let ##h: A \rightarrow A ##be a homomorphism between algebraic structures. Is there a nice result describing the properties of ##Ker h^2 ## , where ##h^2 = hoh ## (composition) ? Clearly , ## ker( h) \subset ker (h^2 )## , but are there some other results relating the two; maybe relating...

I was wondering if there is a field of mathematics which lies beyond and higher than abstract algebra? If it exists could someone tell me the name of that field? Thanks.

Hi, There is a theorem in Abstract which said if g.c.d(x,y)= d (g.c.d the greatest common divisor between x and y) then there exist an integers a,b such that ax + by = d It is a corollary from Euclidean algorithm. Does it has a name ? Thanks in advance.

Which course do you think is more important or interesting to take for someone interested in theoretical computer science or theoretical mathematics, number theory or abstract algebra? I am mainly interested acquiring skills and knowledge that will enable me to prove something significant...

I have a question about abstract algebra so if someone could help me answering this question please ... Suppose P,P' are 3-Sylow subgroup, and let Q be their intersection and N the normalizer of Q. Problem: Explain why is the order of N divisible by 9 ? Thanks for your help. Regards,

Homework Statement Let F be a field with p\inN, a prime natural number. Show that either X^{p}-\alpha is irreducible in F[X] or \alpha has a pth root in F Homework Equations The Attempt at a Solution I'm trying to do this without making reference to the field norm, so far I've...

Homework Statement Let G be an abelian group and let x, y be elements in G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). Assume in addition that (o(x),(o(y)) = 1. Prove that o(xy) = o(x)o(y). The Attempt at a...