What is Abstract algebra: Definition + 458 Threads
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.
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text, and also a continuation of this...
The Math challenge threads have returned!
Rules:
1. You may use google to look for anything except the actual problems themselves (or very close relatives).
2. Do not cite theorems that trivialize the problem you're solving.
3. Do not solve problems that are way below your level. Some problems...
Hello everyone,
I am an International Baccalaureate (IB) student working on my extended essay, which is a mandated 4,000-word research paper. My chosen topic is Quaternions, a mathematical concept I find highly intriguing. The primary aim of my paper is to model the rotation of an asteroid...
picture since the text is a little hard to read
i have no problem showing this is a vector space, but what is meant by complex dimention?
Is it just the number on independant complex numbers, so n?
Let ## \varphi \subseteq A \times B; \psi \subseteq B \times C ##. Then ## \varphi \circ \psi = \left \{ (a, c)| \exists b: (a,b) \in \varphi, (b,c) \in \psi \right \} \subseteq A \times C##.
Task: Let ##\varphi## and ##\psi## are subalgebras of algebras ##A \times B## and ##B \times C##...
...Out of interest am trying to go through the attached notes,
My interest is on the highlighted, i know that in
##\mathbb{z}/\mathbb{6z}## under multiplication we shall have:
##1*1=1##
##5*5=1## am assuming that how they have the ##(\mathbb{z}/\mathbb{6z})^{*}={1,5}## is that correct...
I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way?
In the...
Hello,
I have a question that I would like to ask here.
Let ##L = \left\{ x \in \mathbb{Z}^m : Ax = 0 \text{ mod } p \right\}##, where ##A \in \mathbb{Z}_p^{n \times m}##, ##rank(A) = n##, ## m \geq n## and ##Ax = 0## has ##p^{m-n}## solutions, why is then ##|L/p\mathbb{Z}^m| = p^{m-n}##?
I...
Please, I have a question about automorphism:
Let ##\mathbb{K}## be a field, if ##\operatorname{char}(\mathbb{K})=p ##, then the order of automorphism ##\phi## is ##p##, i.e. ##\phi^p=\operatorname{id}##, where ##i d## is identity map.
Is that right? please, if yes, how we can prove it, and...
I did not use the hint for this problem. Here is my attempt at a proof:
Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##.
Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
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...
Hello,
I am looking for one or more books in combination for self-study of abstract algebra. Desirable would be a good structure of the book with good examples of sentences and definitions. Of course, exercise problems should not be missing.
I am now almost tending to buy the Algebra 0 book by...
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?
Dear Everyone,
What are the strategies from proving a either-or statements? Is there a way for me to write an either-or statement into a standard if-then statements? For example, this exercise is from Dummit and Foote Abstract Algebra 2nd, "Let $x$ be a nilpotent element of the commutative...
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 EquationsThe Attempt at a Solution
I will leave out the initial (1,2,3) part of the permutation. We have S3 =...
One of the last classes I'm taking before finishing my degrees as an undergraduate is abstract algebra. My professor uses the textbook 'Contemporary Abstract Algebra' by Joseph Gallian. The book isn't written terribly nor is the teacher a poor one, but I just find this subject so...
<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...
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...
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...