Abstract algebra Definition and 459 Threads

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with another aspect of the proof of Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows...

Hello! I just start looking at SDG and I'm already having difficulties with a few concepts as expressed by A Kock as: "We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter R" "The geometric line can, as soon as one chooses two...

I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups. Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e, Lgh = gh where g,h ∈ G Which corresponds to a translation by g. Is this an example of a module over a ring...

Let ##M## be a left R-module and ##f:M \to M## an R-endomorphism. Consider this infinite descending sequence of submodules of ##M## ##M \supseteq f(M) \supseteq f^2(M) \supseteq f^3(M) \supseteq \cdots (1)## Can anybody show that the sequence (1) is strictly descending if ##f## is injective...

<Moderator's note: Moved from General Math to Differential Geometry.> Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point. Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...

The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group. What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term? I've been trying to find literature on the groups for the...

Homework Statement For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms. Homework Equations (R,◦), where x◦y=2xy+1 (R*,◦), where x◦y=πxy and R* = R - {0} The Attempt at a Solution For question 1, I found a G2...

Hi, so I have just a small question about cyclic groups. Say I am trying to show that a group is cyclic. If I find that there is more than one element in that group that generates the whole group, is that fine? Essentially what I am asking is that can a cyclic group have more than one generator...

I am trying to learn about free groups(as part of my Bachelor's thesis), and was assigned with Hungerford's Algebra book. Unfortunately, the book uses some aspects from category theory(which I have not learned). If someone has an access to the book and can help me, I would be grateful. First...

How can you prove that there can only be 2 possible four-element group?

I know both are different courses, but what I mean is, will a proof based Linear Algebra course be similar to an Abstract Algebra course in terms of difficulty and proofs, or are the proofs similar? Someone told me that there isn't that much difference between the proofs in Linear or Abstract...

What does "linear" in linear algebra and "abstract" in abstract algebra stands for ? Since I am learning linear algebra, I can guess why linear algebra is called so. In linear algebra, the introductory stuff is all related to solving systems of linear equations of form ##A\bf{X} = \bf{Y}##...

Homework Statement Let ##A,B## be subgroups of a finite abelian group ##G## Show that ##\langle g_1A \rangle \times \langle g_2A \rangle \cong \langle g_1,g_2 \rangle## where ##g_1,g_2 \in B## and ##A \cap B = \{e_G\}## where ##g_1 A, g_2 A \in G/A## (which makes sense since ##G## is abelian...

Hi, I am a math undergraduate major and just finished my first abstract algebra course. Unfortunately, we used the lecturer's notes which are quite dry, without motivation, and it really felt bad. I am really interested in abstract algebra, and thus has decided to re-learn it over the summer...

Homework Statement I can't understand how abstract aljebra helps in creating graphical patterns. I don't find eq related to Groups. Do we consider predefined structures [/B] Homework Equations No equation only patterns. one pattern is attached The Attempt at a Solution I don't know how it...

1. The problem statement, all variables and given/known If each element of a group, G, has order which is a power of p, then the order of G is also a prime power. Homework EquationsThe Attempt at a Solution I am not sure really where to get started. I know that the class equation will be used...

I am looking for an accessible textbook in group theory. The idea here is to use it to learn basic group theory in order to take up Galois Theory. My background includes Calculus I-IV, P/Differential Equations, Discrete Mathematics including some graph theory, Linear algebra, and am currently...

The sum of the $k$ th power of n variables $\sum_{i=1}^{i=n} x_i^k$ is a symmetric polynomial, so it can be written as a sum of the elementary symmetric polynomials. I do know about the Newton's identities, but just with the algorithm of proving the symmetric function theorem, what should we do...

Homework Statement Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic. Homework Equations None The Attempt at a Solution [/B] " => " If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one...

Hello! I am a bit confused about the first Sylow theorem. So it says that if you have a group of order ##p^mn##, with gcd(n,p)=1, you must have a subgroup H of G of order ##p^m##. So, if I have a group G of order ##p^k##, there is only one subgroup of G of order ##p^k## which is G itself. Does...

Homework Statement Note: I'm saying it's very very hard because I still couldn't solve it and I've posted it in stackexchange and no answer till now. I'm posting here the problem statement, all variables and known data in addition to my solving attempts. Because I'm posting an image of my...

So I have two sets, call it ##A## and ##B##. I also have a function ##f:A\rightarrow B##. By themselves, it does not matter (or at the very least make sense) to think of ##A## and ##B## as, say, groups (I'm not really thinking exclusively about groups, just as an example). For that matter, it...

What are good books in universal algebra, given that I have a background in Herstein (Topics in Algebra), Hubbard/Hubbard, Engelking (Topology), and Dugundji (Topology)? I am currently reading Hungerford, and I found a field called universal algebra while searching internet for some concepts...

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What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

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Homework Statement I am supposed to prove or disporve that ##f:\mathbb{R} \rightarrow \mathbb{R}## ##f(x)=\sqrt{x}## is onto. And prove or disprove that it is one to one Homework EquationsThe Attempt at a Solution I know for certain that this function is not onto given the codomain of real...

Hi. I have the following question: Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N? Here is what I think. Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N. Now...

Homework Statement Show that the group (\mathbb Z_4,_{+4}) is isomorphic to (\langle i\rangle,\cdot)? Homework Equations -Group isomorphism The Attempt at a Solution Let \mathbb Z_4=\{0,1,2,3\}. (\mathbb Z_4,_{+4}) can be represented using Cayley's table: \begin{array}{c|lcr} {_{+4}} & 0 &...

Homework Statement Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. Homework Equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...

Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...

Homework Statement If gcd(f(x),g(x)) = 1 and m,n ∈ ℕ, show that gcd(f(x)^m, g(x)^n) = 1. Homework EquationsThe Attempt at a Solution So I had previously proved this for non-polynomials: gcd(a,b)=1 then gcd(a^n,b^n)=1 Proof: a = p1*p2*...*pn b = p1*p2*...*pm then a^n = p1^n*p2^n*...*pn^n...

Homework Statement Suppose a field F has n elements and F=(a_1,a_2,...,a_n). Show that the polynomial w(x)=(x-a_1)(x-a_2)...(x-a_n)+1_F has no roots in F, where 1_f denotes the multiplicative identity in F. Homework EquationsThe Attempt at a Solution Strategy: We have this polynomial...

Homework Statement 1. Let g(x) = x^4+46. a) Factor g(x) completely in ℚ[x]. b) Factor g(x) completely in ℝ[x]. c) Factor g(x) completely in ℂ[x]. 2. Completely factor the given polynomial in ℤ_5. [4]_5 x^3 + [2]_5 x^2 + x + [3]_5 Homework Equations ℚ = {m/n / m and n belong to Z, m is not...

Homework Statement Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2. Homework EquationsThe Attempt at a Solution To determine an irreducible polynomial over F, I think it is sufficient to check the polynomial whether has a root(s) in F, So far, I got...

Homework Statement Let g(x) ∈ ℤ[x] have degree at least 2, and let p be a prime number such that: (i) the leading coefficient of g(x) is not divisible by p. (ii) every other coefficient of g(x) is divisible by p. (iii) the constant term of g(x) is not divisible by p^2. a) Show that if a ∈ ℤ...

Homework Statement Find all real numbers k such that x^2+kx+k is reducible in ℝ[x]. Homework EquationsThe Attempt at a Solution This seems like it is simple, but it is new to me so I am looking for confirmation. We know we can find the roots of a polynomial with b^2-4ab. We want b^2-4ab to be...

Homework Statement Let T:R-> S be a homomorphism of rings. Show that T(0_r) = 0_s. Homework EquationsThe Attempt at a Solution First off, the terminology used is kinda confusing. I take 0_r to be the zero in R. Is this correct? For some reason I recall my teacher quickly saying that it was...

This is a question that came about while I attempting to prove that a simple extension was a splitting field via mutual containment. This isn't actually the problem, however, it seems like the argument I'm using shouldn't be exclusive to my problem. Here is my attempt at convincing myself that...

Homework Statement Let R be a ring and suppose r ∈R such that r^2 = 0. Show that (1+r) has a multiplicative inverse in R. Homework Equations A multiplicative inverse if (1+r)*x = 1 where x is some element in R. The Attempt at a Solution We know we have to use two facts. 1. Multiplicative...

Homework Statement Suppose R is a commutative ring with only a finite number of elements and no zero divisors. Show that R is a field. Homework Equations Unit is an element in R which has a multiplicative inverse. If s∈R with r*s = 1. A zero divisor is an element r∈R such that there exists...

I'm taking an abstract algebra course that uses Hungerford's "An Introduction to Abstract Algebra" 3rd Ed. And while I feel like I'm following the material sufficiently and can do most of the proofs it's hard to learn and practice the material without a solutions guide. How am I supposed to know...

Homework Statement Consider the ring of polynomails in two variables over a field K: R=K[x,y] a)Show the elements x and y are relatively prime b) Show that it is not possible to write 1=p(x,y)x+q(x,y)y with p,q \in R c) Show R is not a principle ideal domain Homework Equations None The...

Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...

Homework Statement Hello all I apologize for the triviality of this: Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this: Find the G.C.D of a= 14+2i and b=21+26i . a,b \in \mathbb{Z} [ i ] - Gaussian Integers Homework Equations None The Attempt...

Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...

Homework Statement Good day all! (p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out) So I was wondering if my understanding of this is correct: The Question asks: "\mathbb{Z}_4 has a subgroup is isomorphic to...

Homework Statement Good day, I need to show: \mathbb{Z}_{4}\oplus \mathbb{Z}_{4} is not isomorphic to \mathbb{Z}_{4}\oplus \mathbb{Z}_{2}\oplus \mathbb{Z}_{2} Homework Equations None The Attempt at a Solution I was given the hint that to look at the elements of order 4 in a group. I know...

Homework Statement Good day all, Im completely stumped on how to show this: |AN|=(|A||N|/A intersect N|) Here: A and N are subgroups in G and N is a normal subgroup. I denote the order on N by |N| Homework Equations [/B] Second Isomorphism TheoremThe Attempt at a Solution Well, I know...

I have a question about Automorphisms. Please check the following statement for validity... An automorphism of a group should map generators to generators. Suppose it didn't, well then the group structure wouldn't be preserved and since automorphisms are homomorphisms this would be a...