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Questions tagged [abstract-algebra]

Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, and modules, among other topics. Whenever you use this tag, please also include related tags like group-theory, ring-theory, modules, etc., in order to clarify which topic of abstract algebra is most related to your question and also to help other users find similar questions through the search engine.

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Roots of a polynomial in an integral domain

Let $R$ be a ring and $f(X) \in R[x]$ be a non-constant polynomial. We know that the number of roots, of $f(X)$ in $R$ has no relation, to its degree if $R$ is not commutative, or commutative but not ...
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3answers
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Nil-Radical equals Jacobson Radical even though not every prime ideal is maximal?

Let's assume we have a commutative ring with identity. Can the Nil-Radical and the Jacobson Radical be equal in a non-trivial case (i.e. not every nonzero prime ideal in said ring is maximal)? Are ...
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1answer
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On the relationship between the commutators of a Lie group and its Lie algebra

I was trying to teach myself some basic Lie theory, and I came across this statement on Mathworld, relating the commutator of a group, $\alpha\beta\alpha^{-1}\beta^{-1}$, to the commutator of its Lie ...
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5answers
7k views

Subgroups of finitely generated groups are not necessarily finitely generated

I was wondering this today, and my algebra professor didn't know the answer. Are subgroups of finitely generated groups finitely generated? I suppose it is necessarily true for finitely generated ...
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2answers
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Upper-triangular matrix is invertible iff its diagonal is invertible: C*-algebra case

Exercise 1.14 of the book Rordam, Larsen and Laustsen "An introduction to K-theory for C*-algebras" asks to prove, that upper triangular matrix with elements from some C*-algebra $A$ is invertible in $...
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1answer
173 views

Alternative, consistent frameworks of mathematics with isomorphic mappings to physical phenomenon

A friend of mine who is quite an aggressive Nominalist told me the other day: "Mathematics and numbers are arbitrary; they can accurately predict physical systems in real life only because they are ...
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1answer
373 views

Find an isomorphism from the octic group $G$ to the group $G'$

Find an isomorphism from the octic group $G$ to the group $G'$: $G = \{e, s, s^2, s^3, b, g, d, t\}$ $$e = (1)\quad s = (1234)\quad s^2= (13)(24)\quad s^3= (1432)\quad b = (14)(23)\quad g = (...
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1answer
502 views

Consider group of permutation matrices and write out elements isomorphic to the group and exhibit it

Consider the group of permutation matrices $G =\{I_3, P_1, P_2, P_3, P_4, P_5\}$ For $n=3$ the permutation matrices are $I_3$ and the five matrices are: \begin{equation*} P_1 = [1,0,0;0,0,1;0,1,0] \\ ...
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3answers
6k views

Elements of alternating group $A_3$

List all the elements of the alternating group $A_3$ written in cyclic notation. I come up with Identity $(1)$ Obviously $(123)$
3
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1answer
691 views

Is there an Artinian module with infinitely generated proper submodule?

Let R be a ring with identity. An R-module M is Artinian if it satisfies the descending chain condition on submodules. What is an example of an Artinian module with a proper submodule that is not ...
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2answers
824 views

Algebraization of integral calculus

It is well known that the differential calculus has a nice algebraization in terms of the differential rings but what about integral calculus? Of course, one sometimes defines an integral in a ...
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4answers
371 views

Show that $\gcd(a,b)=|a| \iff a | b$

I'm reading A Computational Introduction to Number Theory and Algebra, which can be found here as a free download. From the book's exercises, I'm stuck with a proof to show that $\gcd(a,b)=|a| \iff a |...
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2answers
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How do I factorize polynomial over Galois field?

How does the factoring of polynomials over Galois fields work? I cannot seem to understand the basic concept. For example: How do I factorize $x^6 - 1$ over $\operatorname{GF}(3)$? I know that the ...
3
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1answer
884 views

Constructing Non Abelian Groups of Given order

Motivating factor behind this question is the comments which are given for this question asked some hours ago Group of order $105$ So given a group $G$ of order $n$ what are the different methods for ...
3
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1answer
259 views

Centralizer of $M \otimes M$

Let $k$ be a field, $V$ a finite-dimensional $k$-vectorspace and $M \in End(V)$. How can I determine $Z$, the centralizer of $M \otimes M$ in $End(V) \otimes End(V)$? For example, if $$M=[[1,0],[0,...
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6answers
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Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way?

I am of, and I would like to retain, a mindset that mathematics does not have to have numbers as the central object of interest. With that in mind, I have done a fair amount of self-study on topics in ...
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3answers
330 views

Intuition behind tensor expansions of linear maps

Given finite-dimensional vector spaces $V,W$, there is an isomorphism $\text{Hom}(V,W) \rightarrow V^* \otimes W$. In particular, any linear map $\phi : V \rightarrow W$ has a tensor expansion $\sum v^...
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1answer
926 views

The smallest normal set containing a subset $X$, what is a “normal set”?

I've been self studying abstract algebra from these notes. I've ran across the following Lemma: The confusing part is that $X$ is a "any subset" not any "subgroup", and that the term "normal set" ...
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6answers
5k views

Examples of rings with idempotent elements

As a part of my studies in ring theory, I've encountered the concept of an idempotent element, i.e., an element $e$ such that $e^2=e$. Are there some interesting examples of rings with idempotent ...
4
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10answers
817 views

Polynomial satisfying $p(x)=3^{x}$ for $ x \in \mathbb{N}$

Let $p(x)$ be a polynomial with integer coefficients, which is not constant. Then is this condition possible: $$p(x)=3^{x}$$ whenever $x \in \mathbb{N}$. Motivation: https://mathoverflow.net/...
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2answers
373 views

Sequence of Prime ideals in a Polynomial Ring

In the $d$ variable polynomial ring $R=k[x_{1},\cdots,x_{d}]$ show that $0, x_{1}R, (x_{1},x_{2})R, \cdots , (x_{1},x_{2},...,x_{d})R$ is a strictly increasing sequence of prime ideals and there is no ...
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3answers
922 views

Applications of Logic and Algebra in Computer Science

I have used logic for specification of security policies in a security model. One of my reviewers asked a question that "why did you use logic and not algebra for this purpose, and what is your ...
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0answers
107 views

Is there a name for this construction? (taking a pairing and obtaining a perfect one)

I believe that if we are given two modules $M,N$ over a ring $R$, and a pairing between them $M \otimes_R N \to R$, we can construct a perfect pairing $M'\otimes_R N' \to R$ by taking kernels. Is ...
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2answers
909 views

What is the center of $V$, the Klein 4 group?

Please help - in my notes, it is the group $V$ itself. I just want to confirm this. Can you also explain and give an example if that is possible?
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2answers
407 views

Showing $G$ is the product of groups of prime order

Let $G$ be a (not necessarily finite) group with the property that for each subgroup $H$ of $G$, there exists a `retraction' of $G$ to $H$ (that is, a group homomorphism from $G$ to $H$ which is ...
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4answers
1k views

Coloring the faces of a hypercube

I will restate the 3-D version of the problem. In how many ways can you color a regular cube with 2 colors up to a rotational isometry. The answer is of course a special case of Burnsides Lemma which ...
4
votes
2answers
443 views

Number of Elements of order $p$ in $S_{p}$

An exercise from Herstein asks to prove that the number of elements of order $p$, $p$ a prime in $S_{p}$ is $(p-1)!+1$. I would like somebody to help me out on this and also i would like to know ...
4
votes
1answer
171 views

Quantum Jordan Algebras(Still wondering!)

I have attempted to google for this, but the searching is marred by the close relationship of Jordan Algebras and Quantum Mechanics. I have been passively thinking of this question for ages, tonight I ...
4
votes
5answers
827 views

Set-theoretical description of the free product?

There is something in the definition of the free product of two groups that annoys me, and it's this "word" thing: If G and H are groups, a word in G and H is a product of the form $$ s_1 ...
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2answers
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Proofs of the structure theorem for finitely generated modules over a PID

I'm looking for different proofs (references or sketch of main ideas) of the structure theorem for finitely generated modules over a PID. If possible, a comparison in terms of clarity, elegance or ...
3
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1answer
244 views

Zero divisors in $SU_q(2)$

I'm looking at the quantum group $SU_q(2)$ (over ${\mathbb C}$) and can't see why it has no zero divisors. It's clear that $M_q(2)$, the quantum $2 \times 2$ matrices have no zero divisors, but I can'...
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1answer
501 views

Projective modules

I have to prove that if $P$ is a $R$-module , $P$ is projective $\Leftrightarrow$ there is a family $\{x_i\}$ in $P$ and morphisms $f_i\colon P\to R$ such that for all $x\in P$ $$ x= \sum_{i\in I} f_i(...
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2answers
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Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$?

This question is taken from Dummit and Foote (14.9 #6). Any help will be appreciated: Show that if $t$ is transcendental over $\mathbb{Q}$, then $\mathbb{Q}(t,\sqrt{t^3-t})$ is not a purely ...
3
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1answer
142 views

What are the most special spaces between which rigid transformations preserve the structures of the spaces

An affine transformation is a linear transformation followed by a translation. They are morphism between affine spaces. A rigid transformation consists of a rotation and a translation. I was ...
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2answers
455 views

Relationships between Symmetric Groups

Here is a question that I can't find the answer to in my notes or my textbook. Is there any relation between $\text{Sym}_n$ and $\text{Sym}_k$ where $k < n$? Is $\text{Sym}_k \subset \text{Sym}_n$?...
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2answers
137 views

Is there a particular characterization of $Aut(\mathbb{Q}^3)$?

I was considering the group of automorphisms on a vector space $\mathbb{Q}^3$ with the binary operation of addition. Is there a general description of the elements of this group? At first, it seemed ...
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2answers
339 views

Quotienting a Set by an Equivalence Relation such that the Natural Projection is a Homomorphism

When studying group theory, I was interested to learn that a the quotient set $G/N$ of a group $G$ is again a group with $\pi\colon G\to G/N$ a homomorphism when $N$ is a normal subgroup of $G$. Is ...
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2answers
200 views

Reducing Relations to Commutator Relations

I think this is obvious and I am just missing something but: Given some finitely presentable algebra, can we always reduce the relations to commutator relations? And moreover, if yes, then is ...
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2answers
396 views

Can two structures be embeddable in each other, but not isomorphic?

I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for ...
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2answers
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Are localized rings always flat as R-modules?

We know this is true for commutative ring, but if $S\subset R$ is a left and right Ore set, and $S^{-1}R$ its localization by this Ore set, is this always a flat $R$-module?
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3answers
278 views

Can we turn the functor “category ring” into a 2-functor in a natural way?

Let $C$ be a small pre-additive category. Let $R(C)$ denote its category ring, that is, $$ R(C)=\bigoplus_{a,b\in \mathrm{Ob}(C)} C(a,b) $$ as Abelian group, where the direct sum runs over all object $...
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5answers
4k views

How fundamental is the fundamental theorem of algebra?

Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial ...
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1answer
777 views

If the order divides a prime P then the order is P (or 1)

I've just come up with this question as I'm studying for a number theory midterm. If $p$ and $q$ are different prime numbers, and it's known that $2^p \equiv 1 \bmod{q}$, then $q\equiv 1 \bmod{p}$. I'...
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1answer
540 views

Irreducible Polynomials in two complex variables

I am seeking methods to see if a polynomial $f \in O(\mathbb{C^2},0)$ is irreducible. The subject is really new to my and I am studying for myself, for which I don't see about this subject. would ...
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5answers
525 views

Why does $K \leadsto K(X)$ preserve the degree of field extensions?

The following is a problem in an algebra textbook, probably a well-known fact, but I just don't know how to Google it. Let $K/k$ be a finite field extension. Then $K(X)/k(X)$ is also finite with ...
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2answers
6k views

Matrix of Infinite Dimension

Any linear map between two finite-dimensional vector spaces can be represented as a matrix under the bases of the two spaces. But if one or all of the vector spaces is infinite dimensional, is the ...
7
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1answer
449 views

The UFD field lemma

This page contains a result which it refers to as the UFD field lemma. I was wondering if anybody knew of any other references which discuss this result--this page is the only place I've seen it. The ...
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4answers
1k views

Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$

In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)) = f(x) + x$$ (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...
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5answers
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Are all algebraic integers with absolute value 1 roots of unity?

If we have an algebraic number $\alpha$ with (complex) absolute value $1$, it does not follow that $\alpha$ is a root of unity (i.e., that $\alpha^n = 1$ for some $n$). For example, $(3/5 + 4/5 i)$ ...
8
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1answer
436 views

Contravariant Grothendieck Spectral Sequence

I'm currently getting confused about indices in some spectral sequences. Assume we work in the category of modules for simplicity. Let $A^\cdot$ be a (bounded on the right) complex and let $B^\cdot$ (...