Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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$S$ is a non empty set and there are $a$ and $b$ for $c$ and $d$ such that $a\cdot c = d$ and $c\cdot b = d$, prove it is a group

An associative operation $\cdot \ $was defined in $S$ such that $\cdot \ $is associative. Also, for all the pairs $c$ and $d$, there are elements $a$ and $b$ such that: $$a\cdot c = d, \ \ \ \ c\cdot ...
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Verifying if these Cayley tables are from groups

For the first table I noticed that $ab = c \implies abb = cb \implies a = cb$ but in the table, $cb = d$, so this can't be a group For the second table, we have: $ab = c \implies (aa)b = ac ...
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Monomial ideals in $k[x_1,x_2,x_3]$

Let $\mathfrak{a}=\langle x_1^2,x_1x_2^2\rangle$ and $\mathfrak{b}=\langle x_1x_2,x_2x_3\rangle\in k[x_1,x_2,x_3]$. We are to calculate $\mathfrak{a}+\mathfrak{b}$, $\mathfrak{a}\cdot\mathfrak{b}$, ...
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Monomial ideal calculus in many variables

Let $\mathfrak{a}=\langle x_1^3,x_2x_4,x_3^2\rangle$ and $\mathfrak{b}=\langle x_2^2,x_3x_4\rangle\in k[x_1,x_2,x_3,x_4]$. We are to calculate $\mathfrak{a}+\mathfrak{b}$, ...
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Standard notation for indices in group theory?

I've seen three notations for indices in group theory, namely $(G:H)$, $[G:H]$ and $|G:H|$. Is there any of these notations that is standard?
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48 views

Artin's Algebra, Exercise 2.4.11. (1st edition)

I have been working through Artin's algebra book. Here's a simple exercise, but I want to make sure I am not missing anything important. Let $(G,\cdot)$ be a group and let $x,y \in G$ with orders ...
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2answers
39 views

Proof that normalizer and center are subgroups

I've seen this proof for the center of a group $G$: $$C = \{x\in G:xg = gx \ \ \ \forall g \in G\}$$ So, the center is the set of all elements that commute with every $g$ of $G$. This subset of $G$ ...
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84 views

Finiteness of subgroup $\rightarrow$ Finiteness of the group

Let $G$ be a group and $H$ be its abelian and normal subgroup. If $H$ is finite and maximal, prove that $G$ is finite. What I tried : Assume $H=\{e,h_2,\cdots,h_{n}\}$. As for each $j$, we ...
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28 views

Lemma 3.53 Rotman's “An Introduction to Homological Algebra”

In Rotman's book An Introduction to Homological Algebra, in the proof of Lemma 3.53, I think we don't need the fact that $\mathbb Q/\mathbb Z$ is injective, because we don't have $0$'s in the ...
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2answers
182 views

Divisors of zero in polynomial ring

I have the following theorem: McCoy: Let $R$ be a commutative ring with identity. If $f=\sum_{i=0}^na_iX^i$ is a zero divisor in $R[X]$, then there exists a nonzero $c$ in $R$ such that $cf=0$. ...
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Index of every maximal subgroup is prime number

Suppose that finite group $G\neq 1$ and $|G : M| ∈ \mathbb P$ for every maximal subgroup $M$ of $G$. Then prove: $G$ contains a normal maximal subgroup. (we all know that a maximal subgroup is normal ...
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27 views

About sets and algebras.

Let $\mathcal{F}$ be a collection of subsets os some nonempty set $\Omega$. Suppose that $\Omega \in \mathcal{F}$ and that $\mathcal{F}$ is closed under the formation of complements and finite ...
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Ramification in $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$

Let $\pi\ne1+i$ be a prime element of $\mathbb Z[i]$. I am interested in the ramification in the extension $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$, especially over $(1+i)$. I've tried for instance to ...
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Modify this formula : $R/I \cong \phi[R]/\phi[I]$

Let $R$ be a ring and $I$ an ideal of $R$, and let $\phi : R\longrightarrow R'$ be a ring homomorphism. Studying by myself, I have a conjecture the following: $$R/I \cong \phi[R]/\phi[I].$$ This ...
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Iterated square roots over finite field. When do we hit a nonresidue?

Suppose that we are working within the integers modulo $p$ where $p$ is some odd prime number. Suppose that $x_0$ is a (nonzero) quadratic residue mod $p$ then there exists some $x_1$ such that $x_1^2 ...
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Existence of an intermediate field $K\subseteq M \subseteq L$ such that $[L:M]=p$

Let $L/K$ be a finite galois extension (normal and separable). Let $p$ be a prime number which divides $[L:K]$. Is there necessarily an intermediate field $K\subseteq M \subseteq L$ such that ...
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Extending McCoy's theorem to multiple indeterminates [duplicate]

So, working in a commutative ring with unity $R$, I've proven that $f\in R[x]$ is a zero divisor iff there exists $s\in R$ such that $sf=0$. I'm now being asked the followup question to extend ...
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1answer
30 views

Proof that transpose of Hadamard Matrix is also a Hadamard matrix

The question is self explaining from the title, but let me elaborate it. In most of the articles/books I've read, fact that the transpose of Hadamard matrix is also a Hadamard matrix is used, but I ...
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Local Constancy of Rank Function

Recently I asked this question. I believe that I have come up with a solution, but I am unsure, because the proof I have seems too easy to be true, and doesn't make very many assumptions. My ...
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Recommend resources of Abstract Algebra which contain more historical development

I read the Wikipedia page about Abstract Algebra. There is a sentence that says Numerous textbooks in abstract algebra start with axiomatic definitions of various algebraic structures and then ...
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Number of elements of order $2$ in $U(n)$

I have to find out how many elements of $Z/8Z$ that satisfy the equation $x^{2}=1$ Clearly the solutions are the elements of $U(n)$ that have order $2$. Manually I checked ...
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central series of $\frac{G}{Z_2(G)}$

Let $G$ be finite p-group I am trying to make central series for $\frac{G}{Z_2(G)}$ and more inportant what is nilpotency class of $\frac{G}{Z_2(G)}$ since ...
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47 views

Finite normal subgroups of $SO(4)$

What are the finite normal subgroups of $SO(4)$? If these do not exist (or if they are trivial, e.g. from some projection to $SO(2)$), are there different finite normal subgroups of $O(4),$ $U(4)$, ...
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34 views

Presentation of Abelianization of a group

Say $G$ is a finite group with presentation $\langle S | R \rangle$ and let $C$ be the commutator subgroup of $G$. Then $\langle S | R \cup \{ sts^{-1}t^{-1} \} \rangle$ is a presentation of $G/C$. ...
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1answer
62 views

Set of all inner automorphisms is a normal subgroup

In order to prove this, I first proved that the set of all automorphisms from a group $G$ to $G$ form a group under composition: The identity homorphism is an automorphism because sends $x$ from $G$ ...
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assume $ M/N $ be a chief factor of $ G $. Why $ M/N $ has prime order or order $ 4 $?

Let $ G $ is a soluble group and $ \Phi(G) $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal ...
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62 views

Why must two integral domains with $17$ elements be isomorphic?

Why is it true that two integral domains with $17$ elements must be isomorphic to one another? I was studying quotient fields when I saw this question.
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123 views

Rings and Fields

I have a few questions in ring and fields theory. First of all, I was trying to show that the field of quotients of $\frac{\mathbb{Z}_{12}}{\langle 4 \rangle}$ is exactly itself, once it is a field. ...
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minimal normal subgroup of a chief factor of soluble group $ G $ is a minimal normal normal subgroup of $ G $?

Let $ G $ is a soluble group and $ \Phi(G) $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent and $ M \neq 1 $. assume $ M/N $ ...
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need examples of different groups

I need example of different groups having different properties like: class 2 or 3 cyclic commutator cyclic center $Z(G)\le \Phi(G)$ redei group $G=\langle aG',bG' \rangle $ and ... Is there books or ...
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2answers
49 views

homogeneous polynomials over finite fields

Let $F$ be a finite field and $p(X_1,\dots,X_n)\neq 0$ an homogeneous polynomial with coefficients in $F$. Is it possible that $p(x_1,\dots,x_n)=0$ for every $(x_1,\dots, x_n)\in F^n$?
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Polynomial Interpolation and Security

Let polynomial $P$ be $P(x)=g(x).(x−β)$, where $g$ is a polynomial and $\beta \leftarrow \mathbb{F}_p$. We evaluate $P$ at some $\textbf{x}=(x_1,..,x_n)$. This gives us $\textbf{y}=(y_1,..,y_n)$. ...
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Applications and usefulness of universal enveloping algebra

I know the definition of the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, and I know the PBW theorem. My question is the following: Where does the concept of the universal ...
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Is there a method to calculate large number modulo?

Is there a (number theoretic or algebraic) trick to find a large nunber modulo some number? Say I have the number $123456789123$ and I want to find its value modulo some other number, say, ...
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Isomorphism classes of $\mathbb{Z}[i]$ modules.

$\textbf{Question:}$ How many isomorphism classes of $\mathbb{Z}[i]$-modules with exactly $5$ elements are there? $\textbf{My Attempt:}$ Since $\mathbb{Z}[i]$ is a P.I.D and any module with $5$ ...
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Can an element in a Noetherian ring have arbitrarily long factorizations?

Suppose $R$ is a Noetherian ring. Is it possible that an element $r\in R$ have arbitrarily long factorizations? That is, for all $n>0$, is there a factorization $r=a_{1n}a_{2n}\cdots a_{nn}$ such ...
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How to find prime ideals of $3\Bbb Z$?

Which one of these is prime of $3\Bbb Z$? $42\Bbb Z$,$24\Bbb Z$,$12\Bbb Z$,$9\Bbb Z$ and $33\Bbb Z$ I tried to check their factor groups if they are integral domains. because An ideal I in a ...
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Finite fields homomorphism [on hold]

If we know nothing about finite fields, how we can show that there exists homomorphism from ring of rational integers to finite field?
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51 views

Endomorphisms of $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$

Let G be any abelian group, End(G) be the set of all group homomorphisms $\varphi\colon G\to G $. End(G) is a unital ring under the operations + and $\cdot$(Please refer to the link for detail, ...
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Every prime ideal is maximal [duplicate]

Problem: Show that if R is a finite ring, then every prime ideal of R is maximal. My attempt: Let I be a prime ideal of R. Then, by definition of a prime ideal, ab ∈ I implies a ∈ I or b ∈ I for ...
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Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent

Let $ G $ is soluble group with $ \Phi(G) = 1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal ...
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finitely generated ideal and number of generators

let $I$ be a finitely generated ideal of ring $R$. Suppose $I/I^2$ as an $R/I$ module is generated by $r$ elements then question is to prove that $I$ is generated by $r+1$ elements.. I have tried ...
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IS $ F(G) $ a direct product of some minimal normal subgroups of G?

Let $ G $ is a finite group and $ F(G) $ is the fitting subgroup of $ G $. IS $ F(G) $ a direct product of some minimal normal subgroups of G? Why ? $ F(G) $ is the largest nilpotent normal subgroup ...
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Is $Z_2(G)$ abelian?

Let $G$ be non-abelian finite p-group and Let $\frac{Z_2(G)}{Z(G)}$ be elementary abelian group how we can get $Z_2(G)$ is abelian? If it's needed we also know $|Z(G)|=p$ and $|Z_2(G)|=p^3$
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Is $G/H$ always a subgroup of $G$?

Given a normal subgroup $H$ of a finite group $G$, is there always an injective homomorphism $$\varphi:G/H\to G?$$ In other words, is $G/H$ a subgroup of $G$? If we pick an arbitrary ...
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Size of generating set of finite $p$-groups

For a group $X$, let $d(X)$ denote the minimum cardinality of a generating set for $X$. Let $G$ be a non-abelian finite p-group and $H\subset G$ a subgroup with $|H|=p$. Can we say that $d(H)=1$? If G ...
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A questions on the groups by a copy of $\Bbb Z$

Let $G$ be an abelian group and $H$ a subgroup of $G$ such that $G/H$ contains a copy of $\Bbb Z$. Is this true that $G$ contains a copy of $\Bbb Z$? ($\Bbb Z$ is the group of integer numbers)
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Discrete convolution of two sequences

Let $R$ be a commutative ring with unity. A finite sequence $x=\left< x_0,\dots,x_n\right>$ with elements in $R$ is called to be prime if there exists $a_0,\dots,a_n \in R$ such that ...
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1answer
19 views

Symmetric power subspace of tensor product

Consider the symmetric power $S^d(\mathbb{C}^2)$. This can be identified with the space of degree $d$ homogeneous polynomials in $x$ and $y$. Can $S^d(\mathbb{C}^2)$ be considered as a subspace of ...
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Polynomial Interpolating; When $y_i$'s are Changed

This is a comlpementry question to the one posted in: Polynomial Interpolation And polynomial Roots Given $\{(x_1,y_1),...,(x_n,y_n)\}$, we can interpolate a polynomial $P$. Assume polynomial $P$ has ...