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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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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159 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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54 views

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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121 views

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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117 views

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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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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89 views

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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182 views

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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199 views

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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43 views

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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145 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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77 views

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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55 views

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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“Convex” polynomials

Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as: UPDATED (case 0 was missing): 0) $p(x)=x$, i.e., ...
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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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158 views

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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144 views

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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489 views

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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195 views

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 ...
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Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
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Separability of finitely generated projectives over commutative ring

A class $\mathcal{C}$ of $R$-modules is called -separative if $A \oplus A \simeq A \oplus B \simeq B \oplus B$ implies $A \simeq B$ for each $A,B \in \mathcal{C}$ -cancelative if $A \oplus C \simeq ...
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Algebraic Morse theory

In 2005, prof. Emil Skoldberg developed a theory, similar to Forman's Discrete Morse Theory, but suited for arbitrary based chain complexes, in his Morse Theory from an algebraic viewpoint. I'm going ...
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What is the relationship between solvable, nilpotent and transfer homomorphism?

I know some facts such as: Nilpotent groups are solvable, $p$-groups are nilpotent, a finite group whose order is a product of distinct primes is solvable, and finite groups are nilpotent if and only ...
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Examples of exact couples of abelian groups

Exact couples are really important when defining spectral sequences. However, I have never really seen a simple non-trivial example of two exact couples of abelian group with a morphism between them. ...
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321 views

On the Nakayama functor

Let $A$ be a finite dimensional $k$-algebra with 1. Denote by $_AP$ the category of projective left $A$-modules finite dimensional. And with $_AI$ the category of injective left $A$-modules finite ...
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Profinite group power map

I am working through some exercises in Neukirch's Algebraic number theory and i need some hints for exercise 1 pg 274, it goes as follows: Let $G$ be a profinite group, show that we can extend the ...
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Split short exact sequences and the associated graded algebra

Let $G$ be a group and $K$ be a field. Let $K(G)$ be the group ring and $I$ be its augmentation ideal. You are given that there exists a section of the projection $\pi_s:I^s\to I^s/I^{s+1}$ for all ...
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Ext in Dedekind domains

I know and can prove that $\operatorname{Ext}_Z^1(\mathbb{Z}/p\mathbb{Z},A) \simeq A/pA$. Does similar formula work for more general rings, such as Dedekind domains and their ideals, i.e. ...
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241 views

Zero divisors in modules?

Let $R$ be a ring. I find myself considering $M=R^n$, as an $R$-module. If $a$ is not a zero divisor in $R$, it holds that $\forall x \in M: ax=0 \Rightarrow a=0 \vee x=0$ . For what kinds of ...
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Correspondence between summands of a module and primitive idempotents of the endomorphism ring of the module

I am reading this paper and have the following questions. Let $A$ be a finite-dimensional algebra over a fixed field $k$. Let the finitely-generated $A$-module $M$ be a generator–cogenerator for ...
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Noetherian rings, why commutativity?

I am looking for an answer to why one has to assume commutativity of a ring $R$ in proving some results about Noetherian rings. For example, Let $R$ be a commutative ring; look at the proof(s) of the ...
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Curve - non singular curve and its genus

Help me please with this problem:$ X \subset \mathbb{P}^{2}$ defined as $x^{3}y+y^{3}z+z^{3}x=0$ 1.Prove X - non singular curve and find its genus. 2.Prove X - maximal curve over $F_{8}$ field, and ...
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Matrix group as subgroup of $GL(n,\mathbb{R})$ or $GL(n,\mathbb{C}$)?

As far as I understand, one has (at least?) two choices to introduce infinite matrix groups: Either, one can say they are all subgroups of the general linear group over the complex numbers numbers ...
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Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a finite group $G$?

I know that if $H$ and $K$ are subgroups of a finite group $G$, then $|HK|=\frac{|H||K|}{|H\cap K|}$. Is there a formula for $|\langle H\cup K\rangle|$, if $H$ and $K$ are nonempty subsets of a ...
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Homogeneous polynomial in $n$ variables of degree greater than $n(n-1)$ is in the ideal generated by the elementary symmetric polynomials

Let $f(x_1,...,x_n) \in \mathbb{Z}[x_1,...,x_n]$ be homogeneous of degree $d>n(n-1)$, i.e. $f$ is the sum of monomials of degree $d$. I am looking for a hint to prove that $f$ is in the ideal ...
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549 views

What does a Cayley table tell?

Other than showing if a group is abelian and the order of the group, what else does a Cayley table indicate?
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Example of non-Krull integrally closed BFD?

Here's another question in the same spirit as my previous one: Are there any integrally closed BFDs which are not Krull domains? Some background information: A BFD (bounded factorization domain) is ...
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How do I find the orbits given the ring of invariants?

Suppose I have a group $G$ acting linearly on a vector space $V$ and I know the ring of invariants $K[V]^{G}$ (i.e., I know the subring of $K[V]$ which is fixed pointwise under the induced action). My ...
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Is there an integral domain with a lot of residue fields of the same characteristic?

Is there a commutative integral domain $R$ in which: every nonzero prime ideal $Q$ is maximal, and for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such ...