Abstract algebra Definition and 459 Threads

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 > 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 (4)...

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...

Homework Statement define a function f:H--> gHg^{-1} Homework Equations prove if f is 1-1 and onto.The Attempt at a Solution 1-1: f(h1)=f(h2) gh1g^{-1}=gh2g^{-1} h1=h2 (left and right cancellations) onto: f(g^{-1}hg)=gg^{-1}hgg^{-1}=h so every h belonging to H has an image of g^{-1}hg...

Homework Statement Theorem 8.1 of Dan Saracino: Let f ε S_{n}. Then there exist disjoint cycles f_{1},f_{2} .. in S such that f= f_{1}°f_{2}... In proving this theorem, it considers a finite group S_n={1,2,..,n} and chooses x_1 ε S_n. Then it defines x_2= f(x_1), x_3=f(x_2) and so on. The...

I'm trying to round out my math skills in order to apply to graduate school for physics and I've already taken all of the calculus offered along with linear algebra, power series etc... I'm wondering which would be better should I choose to take a math course this term: abstract algebra or set...

1. Show that S42 contains multiple subgroups that are isomorphic to S41. Choose one such subgroup H and find σ1,...,σ42 such that How can you solve this?? I am confused if anyone can help me to solve this!

Homework Statement Let A=C_{p^k} where p is a prime and k>0. Let _{p^m} A consist of all element a of A such that a^{p^m}=e. Prove that _{p^m} A/_{p^m-1} A\cong C_p if m\leq k, \frac{_{p^m} A}{_{p^m-1} A}=e if m>kThe Attempt at a Solution Please could someone explain how to get started with...

Homework Statement Prove that (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is isomorphic to F_4[z]/(z^2 + z + 1) by showing that the kernel of \phi : (\mathbb{Z}/2\mathbb{Z})[x,y] \to (\mathbb{Z}/2\mathbb{Z})[x,y]/(x^2 + xy + x^2, x^2 + x + 1, y^2 + y + 1) is the...

Homework Statement Show via induction that the nth root of (a1 * a2 * a3 * ... an) ≤ 1/ (n) * ∑ ai, where i ranges from 1 to n. Homework Equations Induction The Attempt at a Solution Let Pn be the statement above. It is clear that P1 holds since a1 ≤ a1. Now let us assume that Pn...

Homework Statement Suppose H is a subgroup of G. For g in G, define fg : G/H > G/H by fg (aH) = gaH for a in G, where G/H is the set of left cosets of H in G. I know that fg is a well-defined permutation. However, we have not established (yet) that G/H is a group. 2 parts to the...

Greetings, For a homomorphism \varphi, I'm trying to show that elements of a fiber, say the fiber above a, X_a, are writable as a given element of X_a times an element of the kernel K. So, if a\in X_a and b\in X_a, then \exists k\in K such that b=ak. I want to do this without using the...

Homework Statement The Attempt at a Solution I'm very new to this kind of maths, so don't quite know how to get started. If I understood the question at all we have g_i \mapsto \phi_i and so I have a homomorphism if I can show that \pi(g \cdot g_i) = \pi(g) \circ \pi(g_i) I'm thinking...

Prove that if G is a group and aεG, then o(a-1)=o(a) This is all I have so far: Assume G is a group and aεG. Because G is a group a has an inverse in the group, a-1 s.t. aa-1=e, which is also in G. <a>={an|nεZ}. |<a>| is the number of elements in <a> before it cycles back. Basically all I've...

Could someone try to rank 'Abstract Algebra' textbooks, either undergraduate, or graduate level: By how rigorous they are, how they transfer to applicable subjects, and how well they're laid out, in a pedagogical manner. Any answers would be appreciated. Thanks in advance! SL!

If \ast : (f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d}), show that \ast is commutative. Note that d|n says d divides n. Now I was not sure how to do this from an abstract algebra point of view although when I stare at it my though process was to maybe rewrite it somehow, which will then be...

I am reading at the moment about abstract algebra. It is a very interesting field. I was amazed by the number of examples, applications and related concepts. Never seen something similar in any other mathematical field. I saw lots and lots of theorems and I was wondering whether I should...

Homework Statement Define the set Q[√2] to be the set {a + b√2 | a, b are rationals}, and define addition and multiplication as "usual" (so 2×4 = 8, 2 + 4 = 6, you know, the usual). Show that for any nonzero A in the set Q[√2], there exists an inverse element so that A×A-1 = 1Q[√2]. There...

Homework Statement An interesting example of a ring: Begin with a nonempty set X and form the power set of X, P(X), which is the set of all subsets of X. On P(X), define addition + and multiplication × as follows: For A, B in P(X): A × B = A ∩ B A + B = (A\B) ∪ (B\A), where as...

Hi there, Need one upper div math class to fill out my schedule. It looks like it's a choice between intro to abstract algebra or intro to topology. Which would benefit me more, as a student looking towards grad school?

Abstract Algebra: Relations; Find a relation that is symmetric, etc Homework Statement Find a relation that is symmetric and transitive but not reflexive. Homework Equations None, other than my chosen condition on the relation, namely: xy > |x + y|. The Attempt at a Solution...

Hello. I started Gallians Contemporary Abstract Algebra today. Is it wise to go through each of the given proofs for all of the theorems. For example I just studied the proof for division algorithm. Took Quite some time. I don't know if I could have produced this proof without peeking at the...

If I were to use an abstract algebra book for quick and easy reference which one would it be? Dummit and Foote is very cumulative, is there anything better in the market? And how long would it take to work out all of D + F for an average student with basic background in Algebra?

Homework Statement Problem 35, Section 7.3 of Dummit and Foote: Let I, J, and K be ideals of R. (a) Prove that I(J+K) = IJ+IK and IJ+IK = I(J+K). (b) Prove that if J \subseteq I then I \cap (J + K) = J + (I \cap K). 2. Concern/Question Despite the problem statement specifically...