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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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Algebraic Structures for Logic Design function

am trying to solve exercise questions on FUNDAMENTALS OF SWITCHING THEORY AND LOGIC DESIGN chapter 2 exercise 2.8 Consider binary number (x1 x2 x3 ) and determine the truth-table of a function that ...
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Algebraic Structures for Logic Design

Prove the assignments (Groups, Rings, Fields or Homomorphism) of (x1, x2, x3) for which x1+x2 = x3 and x1 ⊕ x2x3 = 1.
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Center of a group contains a normal subgroup

I'm studying to an exam in abstract algebra, and I'm stuck on the following question. I will appreciate any guidance for this one. Let $G=H\times K$ be a Group and let $N$ be a normal subgroup of G. ...
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A difficulty in understanding the universal property of modules.

The property is given below ( from Dummit & Foote) but I have a difficulty in understanding why it is universal property and what is its importance or when usually we use it? could anyone explain ...
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Ring homomorphisms definition

I'm working on a question where are need to find the number of ring homomorphisms from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}/m\mathbb{Z}$ The previous question involved finding number of group ...
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Tower of fields and quadratic extensions

Suppose there is given a tower of fields in $\mathbb{C}$: $\mathbb{Q} =L_0 \subseteq L_1 \subseteq ... \subseteq L_n \supseteq \mathbb{Q}(\alpha)$ with an $\alpha \in \mathbb{C}$ such that all ...
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1answer
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Action of a 1-form on the push-forward of a vector

I am currently a physicist studying differential geometry I am trying to proof the expression below. Given that for a map $\phi$ : $M$ $\to$ $M$ the pull-back $\psi$*$\omega$ $\in$ $T^\ast_p M$ of a ...
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it is true that if $x^{2}=0$ for any element $x$ of ideal $I$ of ring $R$ then $I\subseteq \operatorname{rad}(R)$?

Does anyone know; is true that if $x^{2}=0$ for any element $x$ of ideal $I$ of ring $R$, then $ I\subseteq \operatorname{rad}(R)$?
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Proof of separability of polynomials without derivatives [duplicate]

Is there a known proof without differentiating that proves that all irreducible polynomials over $\mathbb{Q}$ are separable? (Or even better, for all fields of characteristic $0$.) EDIT: As people ...
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1answer
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Does $D_4$ have a verbal subgroup of order 4?

Does $D_4$ have a verbal subgroup of order 4? How did this question arise: In the comments $Q_8$ ad $D_4$ were pointed to be a possible counterexample to this question: Is it true, that for any two ...
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[Abstract Algebra]A group of questions from a previous exam [on hold]

My girlfriend had an horrendous exam in Abstract Algebra. She worked hard during the semester and spent two weeks studying for this exam day to night (from 8:00 to 22:00) from textbooks and from the ...
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3answers
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Let $N,M$ be normal subgroups of $G$ with $N\cap M=\{e\}$. Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$.

First consider the following definition: Let $G$ be a group and $H$ a subgroup of $G$. The center: $$ C_{G}(H)=\{g\in G\,:\,gh=hg,\,\forall h\in H\}$$ Now I'm trying to prove the following ...
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0answers
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Trace map for extension of local fields

Let $K\supset F$ a finite extension of local fields. It means that the valuation $v_K$ extends the valuation $v_F$. We denote with $\pi_K$ and $\pi_F$ the uniformizer parameters and with $\mathcal O_K$...
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1answer
17 views

Algebras without Nontrivial Subalgebras

Let us define an algebra to be a pair $(A, \mathcal{F})$, where $A$ is a set and $\mathcal{F}$ is a collection of finitary functions on $A$. Some common algebras include groups and rings. I have left ...
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1answer
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Find the distinct left cosets of $S_{n-1}$ in the symmetric group $S_n$.

The number of elements in $S_n$ is $n!$.The number of elements in $S_{n-1}$ is $(n-1)!$. By Lagrange Theorem we have that the number of distinct left cosets of $S_{n-1}$ in $S_n$ is $n$. I assumed ...
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1answer
28 views

${(F^*)}^{-1}(m)$ is maximal ideal where $m$ is maximal

Our main objective is to interpret $F:V \to W$ a morphism as a map $F:maxSpec \Bbb C[V] \to maxSpec \Bbb C[W]$, $V,W$ are algebraic varieties. Now from $F:V \to W$ using Hilbert Nullstelensatz we ...
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1answer
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The existence of a linear map onto an affine algebraic set

Let $K$ be an algebraically closed field, $X\subset K^n$ be an affine algebraic set and $I$ be the ideal generated by all polynomials in $K[x_1,\cdots,x_n]$ that vanish on $X$. Prove that there ...
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1answer
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Find a homomorphism that maps a point in a Boolean algebra into the image of a proper filter

Let $\mathbb B$ be a Boolean algebra. Let $F$ be a proper filter in $\mathbb B$ (i.e. $0 \notin F$), and let $I$ be its dual ideal. Suppose there exists $a \in \mathbb B$ such that $a \notin F \cup I$....
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How to find all submodules of a given $\Bbb R[x]$-module?

In class my professor gave an example where $M$ is the $\mathbb R[x]$-module given by $\mathbb{R^{3}}$ with $x$-action given by $$A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & ...
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Lift $O(\mathbb{Z}/p\mathbb{Z})$ to “something” in $O(\mathbb{Z}/p^2\mathbb{Z})$

So the question was from a result of Serre which basically says if $H$ is a closed subgroup of $Sp_{2n}(\mathbb{Z}_p)$ that maps surjectively onto $Sp_{2n}(\mathbb{Z}/p\mathbb{Z})$, then $H=Sp_{2n}(\...
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1answer
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does this way of grouping numbers have a name?

so this is a system i made, and i was wondering if someone had explored it before, and if it had a name. you start with a list of natrual numbers. you then remove every other number and i will call ...
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2answers
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Find the cycle structure of all the powers of $ (1,2,…,8)$

Fro Topics in algebra Herstein books Find the cycle structure of all the powers of $(1,2,....,8)$? My attempt : i take $T=(1, 2, 3, 4, 5, 6, 7, 8)$ $T^2 = (1, 2, 3, 4, 5, 6, 7, 8)(1, 2, 3, 4, ...
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3answers
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Every two permutation of order $2$ in $S_4$ are conjugate

I was trying to solve the following question: Prove or disprove: every two permutation of order $2$ in $S_4$ are conjugate. I tried to disprove it: $\sigma_{1}=(2,3)$ and $\sigma_{2}=(1,2)(3,4)$ ...
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4answers
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Finding rational solutions for $3x^2+5y^2 =4$

I want to calculate all rational solutions for $3x^2+5y^2 =4$. However, I think that there are no rational solutions because if we homogenize we get $3X^2+5Y^2 =4Z^2$ and mod 3 the only solution is $Z=...
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3answers
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$G$ finite group that has two elements of order $2$ that swap with multiplication [on hold]

Prove or disprove: Let $G$ be a finite group that has two elements of order $2$ that swap with multiplication ($ab=ba$). Then $4$ divides $|G|$. How should I approach it?
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1answer
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If there exists $k$ in group $G$ such that $x^2=kxk$ for any x in G. Show that $(G,\star)$ is Abelian.

If there exists $k$ in group $G$ such that $x^2=kxk$ for any x in G. Show that $(G,\star)$ is Abelian.
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4answers
71 views

Let $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$.

Prove or disprove: let $G$ be a group and $a,b\in G$ elements of order $5$. If $a^3=b^3$ then $a=b$. I saw the following example which tries to disprove the theorem: $G=\mathbb{Z}_{10}$ and $a=2,b=8$....
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1answer
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In a commutative, Noetherian ring, $d(A/J) = d(A/{J^{m}})$

Let $R$ be a commutative, Noetherian ring. Let $d$ be a dimension function: for each finitely generated $R$-module $M$, we assign a natural number, or zero, such that for every $N \leq M$ a submodule ...
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Infinite direct sum of p-adic integers is not p-adic

Studying Bousfield localization I stumbled upon this elementary example: we start with $\mathcal{D}$ the derived category of $p$-local abelian groups and we can consider the Bousfield class of $\...
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3answers
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Each group of order 8 has a subgroup of order 2 and a subgroup of order 4.

So, I was trying to prove the following theorem: Let $G$ be a group of order $8$. So $G$ has a subgroup of order $2$ and a subgroup of order $4$. First I proved that if a group has a finite even ...
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2answers
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Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$.

I came across with the following two theorems: First theorem: Let $G$ be a finite group so $K\leq H\leq G$. Then $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$. Second theorem: Let $G$ be a group so $K\...
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2answers
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Must the morphisms of the category be structure-preserving?I found something different in a textbook.

It is well known that morphisms between the objects of the category are structure-preserving, but I found that in a textbook it said that morphisms are often structure-preserving. Does this mean that ...
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0answers
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Valuation of power of $x \in G$ where $G$ is a complete $p$-valued group

Let $(G, \omega)$ be a complete, $p$-valued group. I want to show that $\omega(x^k) = \omega(x), \; \;\forall x \in G, \; 1 \leq k \leq p-1$ I am trying to derive inequalities using the properties ...
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1answer
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calculating $C_G(a)$ when $a\in G=S_3$

Let $G$ be a group and $a\in G$. $C_G(a)=\{g\in G | ga=ag\}$ is called the center of $a$ in $G$. In order to understand this theorem I'm trying to find $C_G(a)$ for all $a\in G=S_3$. As I understand:...
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Reducing an approximation claim from a prime ideal to a maximal ideal

Theorem. Let $R$ be a domain, $K$ its fraction field and $L$ a field extension of $K$. Let $S$ be the integral closure of $R$ in $L$. Let $P_1,...,P_k$ be prime ideals in $S$ with $P_i \cap R = p$. ...
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1answer
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Is $a*b = a - b + 1$ a binary operation on $\mathbb{Z}^+$?

Possible binary operation is $a*b = a - b + 1$ for all $a,b \in \mathbb{Z}^+$ I don't believe this is a binary operation in any way. A counter example is when $a = 1$ and $b = 5$, then the result of $...
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1answer
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Calculating $G/X$ and $G/Y$ when $G$ is cyclic group of order $10$ and $a$ is the generator of $G$

I came across the following question: Let $G$ be a cyclic group of order $10,$ where $a$ is the generator of $G$. Let $X=\langle a^2\rangle$ and $Y=\langle a^5\rangle$. Calculate the left cosets in ...
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1answer
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Find $3$ non-isomorphic groups of order $2012$

Find $3$ non-isomorphic groups of order $2012$. Is the following correct? First of all, we have the two non-isomorphic abelian groups $\mathbb Z_{2012}$ and $\mathbb Z_{2}\times\mathbb Z_{1006}$. ...
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2answers
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Does the vector space spanned by a Gröbner basis depend on the monomial order?

Let $k$ be a field, $n$ a natural number, and $I$ an ideal of $k[x_0,\dots,x_{n-1}]$. Given a monomial order $M$ on $k[x_0,\dots,x_{n-1}]$ let $G_M$ be the Gröbner basis of $I$ with respect to $M$ and ...
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Necessity of diagonalizability of adjoint representation of cartan subalgebra in definition

This is the definition of cartan subalgebra define in Brian Hall, Lie groups, Lie algebras and representations, 2nd Edition, Chpt 7, Sec 2, Def. 7.10. I am assuming the ground field is $C$ or it does ...
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Question about Lang's Chapter 6 Theorem 9.1

I am an undergraduate working through Chris Hall's result about infinitely many twin irreducible polynomials over finite fields. He begins his argument with a lemma, If $q \equiv 1$ mod $l$ for ...
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0answers
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Extension of bilinear forms

Consider $L/K$ a field extension. Consider $V=K^n$ vector space over $K$ and $B:V\otimes_K V\to K$ $K-$linear map. Now I want to extend the map $B$ by applying $L\otimes_K(-)$ functor to $B$. This ...
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1answer
32 views

How to prove this property of this group of order $20$ without the Sylow theorems?

In Artin's Algebra under the section on the Class Equation is the exercise The class equation of a group $G$ is $1+4+5+5+5$. (a) Does $G$ have a subgroup of order $5$? If so, is it normal? (b) Does ...
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Give an example of a ring with exactly 3 ideals. [on hold]

Give an example of a ring with exactly 3 ideals. (Give a brief explanation.)
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Showing associativity on this binary operation

I am trying to determine whether the definition of * gives a binary operation on the set. On $\mathbb{Z^+}$, define * by letting $a*b = a^b$ I think that the binary operation is not commutative ...
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3answers
51 views

Defining the complex numbers as the algebraic closure of the real numers [on hold]

It is of course possible to define the complex numbers as a quotient ring of real polynomials: https://math.stackexchange.com/a/1083130/359302 But is it possible to prove all of their properties - ...
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Alternate proof for a problem using logic

I applied Modus Tollens and De Morgan's laws to the following property of primes: If $p$ is a prime and $p | ab$ for $a, b \in \mathbb{Z}$, then either $p|a$ or $p|b$. Modus Tollens: If not ($p|a$)...
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0answers
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Valuation of $x^\lambda$ in a complete, $p$-valued group

Suppose for $p$ a prime that $(G,\omega)$ is a complete $p$-valued group, $x \in G$ and $\lambda \in \mathbb Z_p$ (the $p$-adic integers). Let $x^\lambda$ denote the unique element of $G$ such that $\...
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1answer
59 views

Trouble understanding topological groups.

I understand that a topological group is a group $G$ endowed with a topology $\tau$ on $G$ such that addition and inverse are continuous on $\tau$. Now, the definition of continuity is that for all $...
3
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0answers
61 views

Balls and Boxes

Three boxes contain balls. Each box is large enough to contain all balls. We call $\bf{target box}$, a box that receive balls from one of the others boxes. We allow only one process: moving $n$ balls ...