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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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5 views

Why is $k[x_1] \to A$ not finite and $\phi(y)=x_1+x_2$ finite?

I was reading https://www.math.columbia.edu/~dejong/courses/deJongNotes.pdf. See example 1 in the beginning there I could not understand why $k[x_1] \to A$ is not finite[ $A$ is definitely finitely ...
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0answers
7 views

Generators of a Lie algebra

If we know a simple system which corresponds to a Lie algebra, say $\Delta = \left\{ \alpha_1, \alpha_2 \right\}$ for $A_2$ then how do you find a basis for the Lie algebra?
1
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1answer
47 views

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 ...
0
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1answer
19 views

Question on modules of finite length

Let $K$ be a field and let $f$ be a nonzero polynomial on $K$. Then $K[X]/\langle f\rangle$ is a both a $K$-module and a $K[X]$-module. It can be shown that $\ell_{K[X]}(K[X]/\langle f\rangle)\leq \...
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0answers
7 views

Finding the Weyl group from a Cartan matrix

I am looking to establish the relationship between Weyl groups and semisimple Lie algebras. So far I have found the root space decomposition (I am using $\mathfrak{sl}(3, \mathbb{C}) $as my example). ...
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0answers
214 views

**UNSOLVED** Find an integer $\geqslant2$ that is build up out of only $1$'s and $0$'s in base $1,\;\ldots,\;10$.

This riddle bothers me for a few weeks now and I'm starting to worry that I need some $p$-adic Number theory to solve this. I solve most of the riddles in a day, but this one is just annoying to me. I ...
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0answers
17 views

Direct sum and normal sum

What is the differences between $(\hat{e_1}+2\hat{e_2})(e_1+e_2)$ and $(\hat{e_1}\oplus 2\hat{e_2})(e_1+e_2,e_2)\\$ $e_1=(1,0,0), e_2=(0,1,0)$
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0answers
44 views

Proof of separability of polynomials without derivatives

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 ...
4
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5answers
77 views

Subgroup of a finite group H and G

I regret to admit that this has been confusing me for much longer than I would like. I just can't wrap my head around some parts of this question and I'd like some guidance. Let $G$ be a finite group,...
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1answer
40 views

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?and why the function ...
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1answer
22 views

Does a set of all real valued functions form group under componentwise multiplication?

I think I have verified the first two axioms: closure, associativity. For existence of identity, I said that $\forall$ $f$ $\in$ $\mathbb R^\mathbb R$, there is a function $e(x) = 1$ $\in$ $\mathbb R^\...
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0answers
54 views

A set of “pathological” questions from an 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 ...
2
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1answer
30 views

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. ...
0
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1answer
29 views

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 $\phi$*$\omega$ $\in$ $T^\ast_p M$ of a ...
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2answers
48 views

Let $H$ be a subgroup of a group $G$. Show that for $a,b\in G$ we have $aH=bH$ if and only if $a^{-1}b\in H$

My Proof: Suppose $aH = bH$, then $a^{-1}aH=a^{-1}bH$ . . . $e = ab^{-1} \in H$ How to I prove conversely ($\Leftarrow$) say suppose $ab^{-1} \in H$...
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0answers
14 views

Algebraic Structures for Logic Design [on hold]

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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0answers
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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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1answer
98 views

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.
2
votes
1answer
748 views

Galois group for $x^8 - 2$

My textbook asked me to find the Galois group $G$ for $x^8 - 2$. Ok, so the roots of $x^8 - 2$ are $e^{2\pi ik/8} * 2^{1/8}:0 \le k < 8$, by my calculations, so the splitting field is $\Bbb{Q}(2^{1/...
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0answers
22 views

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 ...
3
votes
1answer
33 views

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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0answers
22 views

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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0answers
18 views

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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votes
0answers
23 views

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 & ...
42
votes
3answers
14k views

Finite subgroups of the multiplicative group of a field are cyclic

In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by ...
4
votes
0answers
44 views

Prove that the ring $\mathcal O_K$ of algebraic integers of $K = \Bbb Q (\sqrt d)$ is a Dedekind domain.

Prove that the ring $\mathcal O_K$ of algebraic integers of $K = \Bbb Q (\sqrt d)$ ($d$ is a square free integer) is a Dedekind domain. I have taken an ideal $I \subseteq \mathcal O_K$. Consider the ...
5
votes
4answers
2k views

Show that number of solutions satisfying $x^5=e$ is a multiple of 4?

In a finite group, show that the number of nonidentity elements that satisfy the equation $x^5 = e$ is a multiple of 4. I think that besides $ x, x^2, x^3, x^4 $ also satisfies the given equation but ...
3
votes
0answers
42 views

Does there exist some sort of classification of finite verbally simple groups?

Let’s call a group verbally simple if it does not have any non-trivial verbal subgroup. Does there exist some sort of classification of finite verbally simple groups? $G^n$, with $G$ being a finite ...
2
votes
1answer
49 views

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 ...
3
votes
0answers
62 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 ...
2
votes
1answer
31 views

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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0answers
105 views

Torsion-free abelian groups of finite rank and a subgroup of finite index (Fuchs' problem) - self study

I'm trying to solve the following exercise (Fuchs, "Infinite Abelian Groups", Vol. $2$, pp. $153$, Ex.$5$): "Let $A$ be a torsion-free abelian group of finite rank. If $\phi$ is an isomorphism of $A$ ...
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votes
1answer
73 views
+50

If $K/F$ is normal then $K/I$ is Galois.

Let $K/F$ be a normal extension and $I$ be the inseparable closure of $F$ in $K.$ Let $G=\text{Aut}_F(K),$ i.e., $F$ isomorphisms on $K$and similarly define $H=\text{Aut}_I(K).$ Now I have already ...
1
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1answer
50 views

Extend order of $\mathbb{Q}$ to $\mathbb Q(X)$

I struggle to show that there exists a unique order on $\mathbb Q(X)$ that extends the order on $\mathbb Q$ such that $\forall q \in \mathbb Q, ~~~X>q$. It just means that the order of $X$ is ...
1
vote
3answers
25 views

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
14 views

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 $\mathcal O_K$...
2
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1answer
23 views

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 $a \notin F \cup I$....
0
votes
1answer
29 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 ...
2
votes
1answer
18 views

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 ...
1
vote
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 ...
2
votes
3answers
40 views

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 $\sigma_{2}=(1,2)(3,4)$ ...
2
votes
1answer
84 views

When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$ is short exact sequence of groups. The followings are equivalent: $(1)\ G\cong K \times H;$ $(2)$ The sequence right ...
6
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0answers
40 views

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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votes
2answers
33 views

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
40 views

$G$ 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?
3
votes
4answers
66 views

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=...
1
vote
1answer
29 views

shifted symmetric polynomials

BACKGROUND When defining shifted symmetric polynomials we do it in the following way: Let $\mu=(\mu_1,..., \mu_n)$ be a partition with length less or equal to $n$. Then $$s_{\mu}^*(x_1,...,x_n)=\...
2
votes
1answer
35 views

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:...
2
votes
4answers
74 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 $a=2,b=8$....
6
votes
3answers
330 views

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 ...