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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Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
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Minimal $n$ for embedding of an abelian group into a permutation group $S_n$

Given a finite abelian group $G$, is there a formula or quick algorithm to determine the minimum $N$ so that $G$ can be embedded into the permutation group $S_N$? If $G$ is cyclic of order $n$ I ...
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Unicity inner automorphism symmetric group

I need to prove that the center of the symmetric group $S_n$ with $n\geq 3$ is trivial. I chose to use Lagrange's theorem: bearing in mind that the center is a subgroup, it is easily found that ...
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Number field theoretical analogue of Riemann's Existence Theorem

This question is related to Riemann's Existence Theorem as cited in Neukirch, Schmidt et. al.: Cohomology of Number fields (10.5.1). Let $k$ be a number field, $T \supseteq S \subseteq S_p \cup ...
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Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
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Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for ...
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Extension of Euclidean Domain in which irreducibles have minimal norm

For the ring of polynomials $F[x]$ over a field $F$, there exists a larger ring $\bar{F}[x]$, the ring of polynomials over the closure of $F$, in which irreducibles are linear polynomials -- that is, ...
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galois extension of imaginary quadratic field

Let $K$ be an imaginary quadratic field, and let $K \subset L$ be a Galois extension. Let $\tau$ denote complex conjugation. Show that $L$ is Galois over $\mathbb{Q}$ if and only if $\tau(L)=L$. My ...
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Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
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144 views

Homogenous polynomial and partial derivatives

I'm struggling to understand this part in a book I'm reading: Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog, suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto ...
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67 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
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Systematically describing the Galois Group and Intermediate Fields

In an exercise in the textbook you are asked to describe the Galois Group and the intermediate fields of the extension $$ L=\newcommand{\Q}{\mathbb Q}\Q(\sqrt 2,\sqrt 3)\supset\Q $$ I have noted that ...
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List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...
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104 views

Order of the Normalizer of a Sylow $p$-subgroup in $S_{p}$

Question: Let $p$ be a prime and $P \leq S_{p}$ with $|P| = p$. Prove that $|N_{S_{p}}(P)| = p(p-1)$. I have already solved this problem, but I have come across a proof that is much more elegant than ...
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Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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Application of Sylow Theorems.

I have a group theory exam coming up so I was reviewing through my textbook and I stumbled across a problem that looked interesting: Let $p < q$ be two primes and $G$ be a group of order $pq^3$. ...
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141 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
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153 views

Smallest normal subgroup making quotient abelian, nilpotent, solvable

Given a finite group $G$ that is not abelian, nilpotent, or solvable, what is the smallest normal subgroup $H$ in each case such that $G/H$ is abelian, nilpotent, or solvable (respectively)? In ...
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Calculating a particular direct limit

Suppose we want to compute the direct limit of the direct system $$\mathbb{Z}^{14} \overset{A}{\longrightarrow}\mathbb{Z}^{14} \overset{A}{\longrightarrow}\mathbb{Z}^{14} ...
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Embedding $\mathbb{G}_a$ into $GL_2$

Let $k$ be an algebraically closed field of characteristic $p$. I'd like to find interesting examples of closed embeddings $\mathbb{G}_a(k)\hookrightarrow GL_2(k)$, where $\mathbb{G}_a(k)$ is ...
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Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
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$R$ : a ring with unity, $a^2x=a$. Show that $ax=xa$.

Let $R$ be a ring with unity. For eah $a \in R$, there exists $x \in R$ such that $a^2x=a$. Show that $ax=xa$. I know that $R$ has no nonzero nilpotent elements and $axa=a$. Thus I tried to show ...
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Show $G$ is isomorphic to the multiplicative group $\mathbb{R}^\times$

Let $G=\{x\in\mathbb{R}\mid x>0 \text{ and }x\neq 1 \}$ and define $*$ on $G$ by $a*b=a^{\ln b}$. Show $G$ is isomorphic to the multiplicative group $\mathbb{R}^{\times}$. I need to ...
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168 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
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Module theory and free algebra

I found the injective hull of $k_R$ where $k$ is a field and $R$ is the ring of polynomial with $n$ commutative indeterminants in "Lectures on modules and rings" of T.Y. Lam. I don't know if there ...
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Weyl's Dimension Formula

I'm trying to grasp how Weyl's Dimension Formula works, and I'm having a bit of trouble. As an example, I was trying to calculate the dimension of V($\varepsilon_1$) for gl(3). First, I set the ...
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Finitely Presented Group

Let the descending central series of a group $G$ be $$G=\Gamma^1\geq\Gamma^2\geq\cdots$$ where $\Gamma^{q+1}=[G,\Gamma^q]$ for which $\Gamma^q/\Gamma^{q+1}$ is an Abelian group. Suppose that $G$ is ...
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Orbits of affine group scheme actions

Motivation Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal ...
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torsion subgroup of a finitely generated nilpotent group is finite.

Prove that the torsion subgroup of a finitely generated nilpotent group is finite. More generally, in any group with "almost" no torsion all periodic subgroups are finite. Here "almost" means that ...
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What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$?

A common proof of the simplicity of $A_5$ proceeds as follows. First, note that a normal subgroup is always a union of conjugacy classes. Also, a subgroup has order dividing the size of the group by ...
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Abelian group whose quotient is $p$-divisible

Let $G$ be an abelian group of finite torsion-free rank which has no $p$-divisible subgroup. The group $G$ has a free subgroup $F$ of finite rank and the quotient $G/F$ is $p$-divisible. Can be proved ...
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Structure of a fuzzy subspace

Let $V$ be a vector space over a field $F$ and let $f$ be a function from $V$ to the interval $I:=[0,1]$ satisfying the condition that for any $a \in I$ the set $V_a:=\{v \in V | f(v) \ge a\}$ is a ...
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Showing $\mathbb{Z}_{2} * \mathbb{Z}_{3} \cong\ (a, b\ |\ a^2 = b^3 = e)$

Let $G = (a, b\ |\ a^2 = b^3 = e)$. I recognize there must be an epimorphism $\phi : G \rightarrow \mathbb{Z}_{2} * \mathbb{Z}_{3}$ (the free product) by the Van Dyck theorem, but I must show an ...
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Generation of left ideals in group rings

This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated. Suppose that we are given an (infinite) ...
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What is the intuition of conjugacy classes?

How can I fully understand what are conjugacy classes are in groups? I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
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Is it possible to construct Euclidean function from a Euclidean Domain?

Suppose that we have a Euclidean Domain as a ring, but we are not given a Euclidean function. We know that the ring is a Euclidean Domain. We have an oracle for +, . and whether a/b. Can we construct ...
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Find all the central extensions of $\Bbb{Z}_2$ by $\Bbb{Z}_2 \times \Bbb{Z}_2$.

Find all the central extensions of $\Bbb{Z}_2$ by $\Bbb{Z}_2 \times \Bbb{Z}_2$. At first I was going to say that, since central extensions mean a trivial homomorphism, there must be only one: ...
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151 views

The number of zero divisors

I'm interested whether there are (i'm sure there are!) some facts about how many zero-divisors might be in rings with certain properties. For example is there any connection between the cardinality ...
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Question about comultiplication

I have a question about comultiplication for coalgebras: Suppose $C$ is a coalgebra over the field $k$. How does one show that the comultiplication map $\Delta:C\to C\otimes C$ is a coalgebra map if ...
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Lattices as invertible module

Let $E$ be an etale algebra over $\mathbb{Q}$. In other words, $E$ is a finite sum of number fields. Let $L$ be a lattice in $E$, and $R$ the order associated to $L$. More explicitly, $$R=\{ e\in ...
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Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
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Show that $ord(ab) | \frac{mn}{\gcd(m,n)}$ and $\frac{mn}{\gcd(m,n)^2}|ord(ab)$.

Suppose $G$ is an abelian group. Define $ord(a)=m$ and $ord(b)=n$ where $a,b \in G$. Show that $ord(ab) | \dfrac{mn}{\gcd(m,n)}$ and $\dfrac{mn}{\gcd(m,n)^2}|ord(ab)$. Since ...
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Spectrum of a Laurent polynomial ring

I suspect this question is either easy/known or far too general to answer, but I'm finding it difficult to google for so I'd appreciate directions to good resources on the subject. Can we describe ...
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Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
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Minimal polynomial of a finite purely inseparable field extension

Given $F$, a field with characteristic $p > 0$, and a finite purely inseparable field extension $E$. Then prove that the minimal polynomial $f(x)$ of any $\alpha \in E\backslash F$ will be of the ...
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Classification of $n \times n$ matrix in which each two components differ in each row and column up to automorphism

Suppose, we define a class $A$ of $n \times n$ matrix as follows: $$\text{In each Row }i, \text{for any} j,k\ (1 \leq j,k \leq n )\ a_{ij} \neq a_{ik} $$ $$\text{In each Column }l, \text{for any } ...
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Determining whether or not an extension is a splitting field

On an exam for my modern algebra class, we were asked to determine whether or not $\Bbb{Q}\left(\alpha\right)$ is the splitting field (here using the convention that splitting field means the smallest ...
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An explicit $\Lambda_R^\ell(M)$ when $M$ is not free

Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated. When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation} \Lambda_R^n(M)\cong ...
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Proofs of The Fundamental Theorem of Symmetric Polynomials

I have been considering a few proofs of this theorem, and I noticed that a few of them (for example Proof 1, and Proof 2) prove the theorem first for homogeneous symmetric polynomials and then ...
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Möbius transformations form a simple group

How to show the group $M$ of Möbius transformations is a simple group? I know: $SL_2(\mathbb C)/\{+I,-I\}\cong M$ then if $A \lhd M \implies \phi^{-1}(A) \lhd SL_2(\mathbb C)/\{+I,-I\}$. So if ...