In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.
Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called variety of groups.
I would wish to receive verification for my proof that ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##.
• It is easy to verify that ##A = \{a \in \mathbb{Q}: a^2 \leq 3\} \neq \varnothing##. For instance, ##1 \in \mathbb{Q}, 1^2 \leq 3## whence ##1 \in A##.
• We claim that ##\sqrt{3}## is an...
I would like to show that a LLL-reduced basis satisfies the following property (Reference):
My Idea:
I also have a first approach for the part ##dist(H,b_i) \leq || b_i ||## of the inequality, which I want to present here based on a picture, which is used to explain my thought:
So based...
Hello,
I've been thinking a bit about the definition of the ##i##-th successive minima of a lattice (denoted with ##\lambda_i(\Lambda)##), and I would argue that the ##i##-th successive minimum is at most as large as the largest lattice basis vector ##b_i##.
More formally...
School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑
Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...
let p∈Z a prime how can I show that p is a prime element of Z[√3] if and only if the polynomial x^2−3 is irreducible in Fp[x]? ideas or everything is well accepted :)
I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...
I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces:
##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}##
The definition for the equivalence relation is given in the lecture vidoe as
##(\sum_{j=1}^{J}c_j\psi_j...
I believe that I am correct, the following statement here must be FALSE, right? It has to be false because A union B is like the two entire circles of the Venn diagram and that cannot be a subset of the intersection area, right? Now if this statement was flipped, then it would be true?
My university offers two different two-semester sequences for learning abstract algebra, and I can't decide which one would be better for me, a physics major. Here are the two sequences and their course descriptions, copied and pasted from the university website:
Algebra 1: Theory of groups...
Homework Statement
Find all cosets of the subgroup H in the group G given below. What is the index (G : H)?
H = <(3,2,1)>, G = S3
Homework Equations
The Attempt at a Solution
I will leave out the initial (1,2,3) part of the permutation. We have S3 =...
<Moderator's note: Moved from a technical forum and thus no template. Also re-edited: Please use ## instead of $$.>
If ##R_{1}## and ##R_{2}## are relations on a set S with ##R_{1};R_{2}=I=R_{2};R_{1}##. Then ##R_{1}## and ##R_{2}## are bijective maps
##R_{1};R_{2}## is a composition of two...
This time my struggle is with ring ideals. Book still won't provide examples, so I'm again trying to come up with some of my own. I figured {0,2} might fit the definition as an ideal of ##\mathbb{Z/4Z}## since it is an additive subgroup and ##\forall x \in I, \forall r \in R: x\cdot r, r\cdot x...
So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following:
Consider the Klein four group, if we...
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...
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...
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...
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Thread
abstractalgebra
group theory
isomorphism
proof verification
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...
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 Equations
The 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
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
Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2.
Homework Equations
The 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...
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...
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 Theorem
The Attempt at a Solution
Well, I...
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...
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...