The abstract-algebra tag has no wiki summary.

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### What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...

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14 views

### Decompose a multivariate polynomial into a permutation on $F_{2^n}$ and an affine transformation

Let $S(x_1,...,x_n)=(y_1,...,y_n)$ be a secret permutation on $F_{2^n}$. $L$ is a secret $F_{2^n}\rightarrow R^{m}$ affine tranformation. $m$ can be smaller than $n$, while $n$ is ususally less than ...

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### Proving the algebraic independence of certain elements

Let $k$ be a field of characteristic zero and $R$ be the polynomial ring $k[x_1,...,x_n,t_1,...,t_n]$. Let $P_i = (a_{i1}:a_{i2}:a_{i3})$ be $n$ points in the projective plane over $k$, such that not ...

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**1**answer

64 views

### Maximal group image of the Clifford semigroup?

How to prove that if S is a finitely generated Clifford semigroup its maximal group image is actually the S_{e_{n}}?
Or equivalent: how to prove is S is a finitely generated Clifford semigroup then it ...

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**1**answer

93 views

### How to check if argument of complex number is rational multiple of $\pi$ [closed]

Assume there is give a complex number $z\in \mathbb{C}$ of modul 1. Therefore it can written as $e^{i2\pi x}$. What are the known criteria for deciding if $x$ is a rational number???

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### Categorizing finitely presented groups by complexity

Let $G$ be a finitely presented group tagged with a specific presentation $G \cong F\{X\} / \{h_1, \dots h_k\}$.
Preliminary: consider two presentations to have the same presentation class if they ...

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**1**answer

262 views

### Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...

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**2**answers

225 views

### Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic.
Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$.
Is there some ...

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### Do convolution and multiplication satisfy any nontrivial algebraic identities?

For (suitable) real- or complex-valued functions f and g on a (suitable) abelian group G, we have two bilinear operations: multiplication -
(f.g)(x) = f(x)g(x),
and convolution -
(f*g)(x) = ...

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**2**answers

165 views

### Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...

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### What is the diameter of the Cayley graph of $S_n$ w.r.t the generator $\{(12),(23), . . . ,(n − 1 n) \}$? [duplicate]

Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of ...

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### Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...

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289 views

### Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin.
Now, for self dual ...

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### Separability of a simple ring extension

Assume $A=K[x,y]\subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...

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**1**answer

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### How can the Cayley-table for the elements of basis of a Cayley-Dickson algebra be summarized in an algebraic expression?

One would be able to construct a Cayley table that has all $e_i$ elements of the basis of algebra $A$ where $0<i<\dim A$ such that $e_0=1$ and $e_1=i$ and $e_2=j$ and so on. I'm looking for an ...

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**1**answer

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### Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...

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166 views

### What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?

Definition 0. Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the ...

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### Classification of symplectic representations of quaternion division algebras

I would like to know the classification of representations of the form $\rho:B^{\times}\to Sp(V,F)$ or ($Gsp(V)$), where $B$ is a quaternion division algebra over a number field $F$ (or ...

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563 views

### Stanley-Reisner ring of a simplicial complex is a functor?

Let $K$ bea field and $[n]=\{1,\ldots,n\}$ and $K[x]=K[x_1,\ldots,x_n]$. For $\sigma=\{i_1,\ldots,i_k\}\subseteq [n]$, denote $x_\sigma=x_{i_1}\cdots x_{i_k}=\prod_{i\in\sigma}x_i\in K[x]$. Let ...

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### Is there any research of universal algebras axiomatized by non-Horn clauses?

A Horn clause in the language of a universal algebra is a disjunction of equations and of at most one inequality
("equation" and "inequality" are the terms used by A.Horn in his paper "On sencences ...

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### Does there exist a continuous surjection? [closed]

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...

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### Can you “combine” Ord and Mon to get Cat?

Mon is the category of moniods, which can be seen as categories with one object. Ord is the category of preorders, which can be seen as categories with up to one morphism in each homset.
Is there ...

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**1**answer

169 views

### Why do we not lose any generality by proving it only for finitely generated groups [closed]

In the proof of following theorem, in a paper by Farkas-
Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...

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526 views

### Obstructions for a group to be the multiplicative group of a field [duplicate]

It is well known that every finite multiplicative subgroup of a field is cyclic.
I somehow got interested in a possible reverse implication:
Assume we have an abelian group $G$ whose every finite ...

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**7**answers

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### Consequences of not requiring ring homomorphisms to be unital?

As defined in many modern algebra books, a homomorphism of unital rings must preserve the unit elements: $f(1_R)=1_S$. But there has been a minority who do not require this, one prominent example ...

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### the direct sum of injective modules need not be injective

The Bass-Papp Theorem asserts that a commutative ring $R$ is Noetherian iff every direct sum of injective $R$-modules is injective. Thus every non-Noetherian ring carries a counterexample.
If
$$
I_1 ...

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### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

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### Solvable Lie algebras: embedded in upper triangular matrices?

Let $K$ be an arbitrary field and $\mathfrak{g}$ a finite-dimensional Lie $K$-algebra.
Let $\mathfrak{nil}_n\leq\mathfrak{sol}_n\leq\mathfrak{gl}_n$ be the Lie algebras of all ((strictly) ...

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### Counting zero-sum free sequences of a given length in $\mathbb{Z}_n$

Let $n$ and $d$ be positive integers. Define $\alpha_n^d$ to be the number of vectors $(x_1, x_2, \cdots, x_d)$ in $\mathbb{Z}_n^d$ such that given any subset $S$ of $\{ 1, 2, 3, \cdots d\}$, ...

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**1**answer

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### The automorphism groups of smallest grammars of a language string are isomorphic

Let $s \in \Sigma^*$ be a formal language string. Consider the automorphism group of $s$, defined to be the set of all permutations of positions of $s$ that leave $s$ fixed. For instance $G(abab) = ...

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**1**answer

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### When does a faithful module have an element with zero annihilator?

This is a follow up of Example of a finitely generated faithful torsion module over a commutative ring.
Let $M$ be a finitely generated module over a commutative ring $R$ with the property that ...

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### What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null

Let $\mathbb F_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces ...

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### Useless math that became useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.
My idea is to amend my article with some theories that seemed useless when they are created but ...

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### What kind of algebra is the class of ordered pairs equipped with the binary operation which forms them?

There are many definitions of ordered pair in set theory, but all such definitions have the characteristic property of ordered pair:
$ \ \ \ \ \ \ (x, y) = (x', y') \leftrightarrow \ (x = x' \ and \ ...

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### Can we give efficiently the solution of a bilinear system of equations over a finite field?

Consider a finite field $F$ and suppose we have a system of equations
$$h_1(\alpha,\beta)=0,h_2(\alpha,\beta)=0,...,h_t(\alpha,\beta)=0$$
where $\alpha=(\alpha_1,...,\alpha_s)$ and ...

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### Making an algebra the uniqe maximal one-sided ideal in another unital algebra

If $R$ is an algebra without a unit, then the standard unitisation $R^\sharp$ can have maximal one-sided ideals other than $R$. Thus, it is natural to ask about the following. Let $R$ be an algebra ...

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**1**answer

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### An example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$

In the wake of my curiosity on this kind of things, I was thinking if there is an example of a non-(locally cyclic) Abelian group whose automorphism group is cyclic not of order $2$. Every example I ...

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### Example of a Group which has $\text{SL}_{n}(\mathbb{Z})$ as the automorphism group

For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me:
What is the Autmorphism group of ...

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**1**answer

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### A group whose automorphism group is cyclic

Is there an Abelian group $A$ which is not locally cyclic whose automorphism group is cyclic ?
This question was first posted here.

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### About alternating polynomials an the Rowen's notation

Some definitions...
Definition 1: A polynomial $f(X_1,\dots ,X_d)$ is $t$-linear if the variables $X_1,\dots ,X_t,\; t\leq d$ appear in all monomials of $f$ and degree of $X_i,\; i=1,2,\dots ,t$ on ...

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### Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...

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### How do you decide whether a question in abstract algebra is worth studying?

Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my ...

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### What are the reasons for considering rings without identity?

I think a major reason is because Lie algebras don't have an identity, but I'm not really sure.

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### TM and abstract algebra

Usually, during lectures Turing Machines are firstly introduced from an informal point of view (for example, in this way: http://en.wikipedia.org/wiki/Turing_machine#Informal_description) and then ...

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### Is the “algebraic closure” of the quaternions, finite dimensional? [closed]

This post is a sequel of: What's the algebraic closure of the quaternions?
$\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$, but it is not for the polynomials ...

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### Polynomiality of functions over residue rings

Suppose $\mathbb{Z}/m \mathbb{Z}$ is a residue ring for some $m \in \mathbb{N}$. If $m=p$ is a prime number then every function $f:\mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p \mathbb{Z}$ is a ...

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### A More Advanced Version of Aluffi's Chapter 0

This is a crosspost of this question from MSE.
Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce ...

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### motivation of filtered colimits

I am trying to move in categorical algebra beyond the basics. A Lawvere theory L is a small category with finite products. (I know that there also is a functor $(skeleton(FinSet))^{op}\to L$, which ...

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**1**answer

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### Action of the homotopy braid groups on reduced free groups

Firstly some definitions:
$B_n$ is the braid group with $n$ strands.
$\widetilde{B_n}$ is "homotopy braid group", which is a factor group of $B_n$ by adding the relation that $A_{j,k}$ ...

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### Algebraic dependency over $\mathbb{F}_{2}$

Let $f_{1},f_{2},\ldots,f_{n}$ be $n$ polynomials in $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
such that $\forall a=(a_1,a_2,\ldots,a_n)\in\mathbb{F}_{2}^{n}$ we have $\forall ...