# abstract algebra Definition and Topics - 78 Discussions

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.

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1. ### I Irreducible polynomials and prime elements

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 :)
2. ### A What is the geometric approach to mathematical research?

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...
3. ### A Equivalence Relation to define the tensor product of Hilbert spaces

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...
4. ### I This would be a false statement, correct?

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?
5. ### Courses Physics and Abstract Algebra

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...
6. ### Finding Cosets of subgroup <(3,2,1)> of G = S3

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 =...
7. ### How can I prove that these relations are bijective maps?

<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...
8. ### I Polynomial ideals

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...
9. ### I Concerning Quotient Groups

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...
10. ### A Number Line in Synthetic differential geometry

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...
11. ### I Rings, Modules and the Lie Bracket

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...
12. ### A A decreasing sequence of images of an endomorphisme

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...
13. ### I Proving that an action is transitive in the orbits

<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...
14. ### I What are the groups for NxNxN puzzle cubes called?

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...
15. ### Determining a group, by checking the group axioms

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...
16. L

### I How many generators can a cyclic group have by definition?

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...
17. ### I Free Groups

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...
18. A

### I How can there only be two possible four-element groups?

How can you prove that there can only be 2 possible four-element group?
19. ### B What do "linear" and "abstract" stand for?

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}##...
20. M

### Show isomorphism under specific conditions

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...
21. ### Algebra Question Regarding Purchasing an Algebra Book

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...
22. ### I Express power sums in terms of elementary symmetric function

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...
23. ### Group is a union of proper subgroups iff. it is non-cyclic

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...
24. ### I First Sylow Theorem

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...
25. ### A very very hard college algebra problem

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...
26. ### B Sets and functions that gain more structure with context

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...
27. ### Algebra Books for Universal Algebra?

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...
28. ### Insights Comments - How to self-study algebra. Part II: Abstract Algebra - Comments

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29. ### I Group theory

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.
30. ### Schools In High School and Want to Do Advanced Mathematics? - Comments

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