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Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

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0answers
20 views

Divisibility in Integers and General Rings [duplicate]

If x|y and y|x in integers, does this imply x=y? Similarly, for any general ring, if x|y and y|x in the ring, does this imply x=y?
2
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2answers
59 views

Is $D_n$ isomorphic to $C_n \times C_2$?

I am doing a proof for an exercise where if the dihedral group $D_{n}$ $\cong$ $C_{n} \cdot C_{2}$ where $\cong$ means isomorphic and $\cdot$ is the direct product, I would be able to finish it, but I ...
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0answers
18 views

Growth rate of free nilpotent group of rank $r$ and nilpotency index $s$.

Let $F^{(r)}$ denote the free group of rank $r$ with generators $x_1, \dots, x_r$. Recall a group $G$ is nilpotent of index $s$ if $G_{s+1} = \{e\} $ and $G_s \neq \{e\}$ (where $G_i$ denotes the ...
2
votes
2answers
62 views

Factorization of a polynomial in $\Bbb F_7$

I need to reduce as much as possible the polynomial: $x^6+3x^5+2x^4+6x^3+4x^2+5x+2$ in the finite field $\Bbb F_7$. It has no roots over the field, and I don't think that it is necessary to check ...
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0answers
27 views

A question regarding the Galois extension of the cyclotomic polynomial $\Phi_{15}$

Given the cyclotomic polynomial $\Phi_{15}$, I am trying to : i) Determine the isomorphism type of the Galois group of $\Phi_{15}$ over $\Bbb Q$. ii)Letting ω be a primitive 15-th root of unity in $\...
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0answers
21 views

Double centralizer theorem for semisimple rings

Why every semisimple ring has the double centralizer property? Namely, let $R$ be a semisimple ring. We consider $R$ as left $R$-module. Let $E=End(R)$. Why every element in $End(R_E)$ is induced by ...
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1answer
24 views

Question about conjugacy classes in groups

I'm trying to prove that the conjugacy class ($\overline{a}=\left\{ b\in G/\exists x\in G\,b=x^{-1}ax\right\} $) in a group satisfy that it has only $a$ as element if and only if $a\in Z\left( G\right)...
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1answer
38 views

Matrix representation of Maschke's theorem

I know Maschke's Theorem in the following form: Let $\ G$ be a finite group over $\ F$ and let $\ V$ be an $\ FG$-module. If $\ U$ is an $\ FG$-submodule of $\ V$, then there is an $\ FG$-submodule $...
2
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0answers
51 views

Can we make a square from a rectangular sheet of paper by folding it into halves or thirds breadth-wise and length-wise? [on hold]

Let we have a rectangle shaped paper sheet. Suppose we have 4 rules that we can apply on that paper. The rules are: We can fold the paper breadth wise 3 times.( i.e if the length of the paper is $l$ ...
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1answer
34 views

Why can't modules $M$ in a stable category $\underline{\operatorname{Mod}}\Lambda$ have non-trivial projective summands?

I'm working through some results in Almost split sequences in subcategories by M. Auslander and S. Smalø, specifically the results 5.6 - 5.10. The specific result I'm trying to understand is ...
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0answers
7 views

Orthogonal idempotent, Dunkl operators

I would like to check te following: Let $V$ be a $\mathbb{C}$-vector space of dimension ($n$); let $G$ be a complex reflection group (see BMR); let $\mathcal{A}$ the hyperplan arrangemant of ...
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0answers
21 views

Intersection with direct sum of modules [on hold]

Let $R$ be an arbitrary ring and $M=\bigoplus_{i\in I} M_{i}$ be a direct sum of $R$-modules. Give an example shows that for a submodule $N$ of $M$ it is not necessary that $N=\bigoplus_{i\in I}N\...
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2answers
183 views

What is the Characteristic of a local ring? [on hold]

What is the Characteristic of a local ring ? We define Characteristic of a Commutative ring with $1$ say, $A$ in the following way: Define a ring homomorphism $\phi: \mathbb{Z} \to A$ by $\phi(n)=n \...
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1answer
26 views

Tensor product of module homomorphisms is element of tensor product or mapping?

Here: $A$ is a ring. $E,E'$ are right $A$-modules, $F,F'$ are left $A$-modules. $u:E\rightarrow E'$ and $v:F\rightarrow F'$ are $A$-module homomorphisms. With this setup, I didn't understand ...
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1answer
26 views

Property of Jacobson Radical

I need to show that: For a ring R, if R has no left primitive ideals, then J(R) = R , where J(R) is the Jacobson radical of R. I have tried in the following way: Let R have no primitive ideals and J(...
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1answer
26 views

Tensor product of modules: should it be abelian group or module?

If $A$ is a right $R$-module and $B$ is a left $R$-module, then the tensor product $A\otimes_R B$ is an object $X$ with a map $\theta:A\times B\rightarrow X$ such that $\theta$ is $R$-bilinear and $\...
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0answers
50 views

Group is isomorphic to opposite group. [on hold]

Given a group G, define a new group G^op having the same set of elements, but a new binary operation ⊙ given by a ⊙ b = ba (where, on the right, we are computing the product of b and a, in that order, ...
2
votes
2answers
51 views

Showing that $[G:K]=[G:H][H:K]$ for $K \leq H \leq G$ and $|G| < \infty$

Definition: For a finite group $G$ and a subgroup $H \leq G$, $$[G:H] := \frac{|G|}{|H|},$$ which is a positive integer. For a finite group $G$, assume $K$ is a subgroup of $H$ and $H$ is a ...
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1answer
36 views

Why is this map surjective?

The author proves that there are two isomorphism classes of groups of order 21: the class of the cyclic group $C_{21}$, and the class of a group $G$ generated by two elements $x$ and $y$ that satisfy ...
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0answers
51 views
+50

Prime ideals of $R[x]$ that intersect $R$ in $P$

Let $R$ be a noetherian ring and $P$ a prime ideal of height $h$. Show that the prime ideals $Q\subset R[x]$ that intersect $R$ in $P$ are of the following two kinds, with height as shown: $Q=...
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0answers
23 views

Characters table of non abelian group of order 21

I am trying to understand the Characters table of non abelian group G of order 21. In the attached picture, I understand how we get the 3 first irreducible representations of dimension 1. I guess we ...
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1answer
38 views

Find the Kernel of $\Psi : (\Bbb{Z}_{30},+_{30})\rightarrow (\Bbb{Z}_{20},+_{20})$

Find the Kernel of $\Psi : (\Bbb{Z}_{30},+_{30})\rightarrow (\Bbb{Z}_{20},+_{20})$ Provided that $\Psi([b])=[4b]$ My understanding of the kernel is that it should be: $\ker(\Psi)=\{[b]\in\Bbb{Z}_{...
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0answers
21 views

Bounding the exponent in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_0$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_1:=1+\operatorname{rad}(KG)$ is a p-group ...
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0answers
24 views

Let $A_1$, $A_2$ be groups and $\phi:A_1\to A_2$ surjective with $G_1\unlhd A_1$ and $\phi(G_1)=G_2$. Is $A_1/G_1\cong A_2/G_2$? [duplicate]

Let $A_1$ and $A_2$ be a group and $\phi: A_1 \to A_2$ be a surjective group homomorphism. Also, $G_1 \unlhd A_1$ and $\phi(G_1) = G_2$. I have to prove/disprove this statement: Is $A_1/G_1 \cong A_2/...
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0answers
24 views

Clarification on normality and separability in the context of field extensions. [duplicate]

I'm a little confused in regards to the definitions of normality and separability of field extensions in the context of Galois theory . The definitions seem very similar . In class they were defined ...
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0answers
43 views

Show that finite subgroup of units of a field is cyclic.

Let $F$ be a field. Show that every finite subgroup of $F^\times$ is cyclic. My attempt: Let $H$ be a subgroup of $F^\times$. Suppose $p\mid|H|$ with $p$ prime. Any element in $H$ of order $p$ ...
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0answers
16 views

Indecomposable G-equivariant vector bundles

I am reading the paper "The representation ring of the quantum double of a finite group" by Witherspoon. In Chapter 2 we define a G-equivariant vector bundle on a finite G-set $X$, as a collection of ...
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3answers
56 views

Given a group $H\unlhd G$, is it natural/accurate to say that $G \cong H \oplus (G / H)?$ [duplicate]

Given a group $G$ and a normal subgroup $H \leq G$ , is it natural / accurate to say that $$G \cong H \oplus (G / H)$$ This is what I feel is true intuitively, but I'm sure a group theorist could ...
2
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1answer
11 views

Projection map for polynomial rings

Let $K$ be a field and Consider the projection map $\pi_{i,j} : K[X]/(X^i) \to K[X]/(X^j)$, for $j \leq i$. This is well-defined since $(X^i) \subseteq (X^j)$. I'm wondering what it looks like, is it ...
0
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1answer
47 views

Irreducible and connected components of $\mathbb{C}[x,y]/(xy)$

There is a question which states that $\operatorname{Spec}(\mathbb{C}[x,y]/(xy))$ consists of three prime ideals: $0, (x), (y)$ I want to find irreducible and connected components of $\operatorname{...
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4answers
43 views

Define dimension without referring to bases

The dimension of a vector space is the common cardinality of all bases. I would like to define it in a way that does not refer to bases. I only care about finite dimensional vector spaces. First, I ...
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1answer
17 views

Tower of ring extensions and flatness

Consider the following ring extensions: $$A\hookrightarrow\ B\hookrightarrow C$$ where we know that: $A$ is a complete DVR, $B$ and $C$ are Noetherian, Two-dimensional local rings and $B\...
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1answer
32 views

How does the ring of algebraic integers in $\mathbb{Q}[\sqrt{D}]$ depend on $D\mod 4$?

Let $O_K$ be the ring of algebraic integers inside of $\mathbb{Q}[\sqrt{D}]$. Why is it that when $D \equiv 2, 3 \mod 4$ that $O_K = \{x + y\sqrt{D} : x, y \in \mathbb{Z}\}$ and that when $D \equiv 1 \...
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0answers
25 views

Product of characters that is not irreducible

I am looking for an example of product of irreducible characters that is not irreducible. I tried the character table of $S_4$ and $D_4$ and thought about $\Bbb Z/6\Bbb Z$ but I couldn't find such ...
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0answers
44 views

Quaternion group of order 12 [on hold]

What are the elements of the quaternion group of order 12 ? I found this on wikipedia but I don't know what are their elements.
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2answers
39 views

Equivalence between these tensor product definitions

Let $V$ and $W$ be vector spaces. Then the tensor product $V \otimes W$ of $V$ and $W$ is the vector space $V \otimes W$ together with a bilinear map $\phi: V \times W \rightarrow V \otimes W$ such ...
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0answers
24 views

Tower of Splitting Fields

If we have $F_n\geq\ldots\geq F_1$ fields such that $F_{i+1}$ is a separable splitting field over $F_i$ for all $i=1,\ldots,n-1$ is it true that $F_n$ is a splitting field over $F_1$? I think I can ...
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1answer
15 views

A particular complex of integral group ring is exact: proof of Jacobson

Let $A$ be a $G$-module. Let $C_n=\mathbb{Z}G\otimes \cdots \otimes \mathbb{Z}G$ ($n+1$ copies). It is free $\mathbb{Z}$-module with basis $g_0\otimes g_1\otimes \cdots \otimes g_n$, $g_i\in G$. ...
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0answers
14 views

Baer sum, pushback of pushout and pushout of pullback

Let us consider the following constructions in the category of $R-$modules, for some ring $R$. Given a short exact sequence $$ \mathcal{S}: \quad 0 \to A \overset{\alpha}{\to} B \overset{\beta}{\to} ...
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0answers
14 views

Is this proof of a irreducibility criterion in an integral domain correct?

This is an exercise from Grillet's "Abstract Algebra" (page $145$, proposition $10.10$). Let $R$ be an integral domain, let $I$ be an ideal of $R$, and let $\pi\colon R\to R/I$ be a canonical ...
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0answers
25 views

List of extension theorems

As a post grad student, I have come across many results where a function with certain properties(eg-homomorphism) on a smaller algebraic structure is extended to a larger one. For example, extending ...
6
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2answers
245 views

Computing $\phi(\frac32)$ where $\phi$ is an automorphism of $\mathbb Q[\sqrt2]$ such that $\phi(1)=1$ and $\phi(\sqrt2)=\sqrt2$

This question is a followup to this question about Field Automorphisms of $\mathbb{Q}[\sqrt{2}]$. Since $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$ with basis $\{1, \sqrt{2}\}$, I ...
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1answer
62 views

Show that there exists an $\alpha \in K$ such that $\alpha^2 \in F$ but $\alpha \notin F$

Let G be a group of order $2^n$ and suppose that $G=Gal(K/F)$ where $F \subseteq K$ is a Galois, separable, normal extension. Then show that there exists an $\alpha \in K$ such that $\alpha^2 \in ...
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2answers
52 views

minimal polynomial for $\sqrt{-3}+\sqrt{2}$ over $\mathbb{Q}$

Almost have the answer. Let $a = \sqrt{-3}+\sqrt{2} \implies (\sqrt{-3}+\sqrt{2})(\sqrt{2}-\sqrt{-3}) = 5 \therefore $ a is a root of $x^4+2x^2+25$. $\sqrt{2} = \frac{1}{2}(a+5/a) \in \mathbb{Q}(a), \...
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2answers
35 views

Short exact sequence of algebras implies bimodule structure

I read a statement in "Algebraic Operads" which I don't understand : Let $$0 \rightarrow M \rightarrow A' \rightarrow A \rightarrow 0 $$ be a short exact sequence of associative algebras over the ...
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votes
1answer
24 views

On natural homomorphism $\nu:R\longrightarrow S^{-1}R$

Let $R$ be a commutative ring with $1_R$ and $S$ an multiplicatively closed set. We define the natural homomorphism \begin{align*} \nu:R &\longrightarrow S^{-1}R, \\ a&\longmapsto \nu(a):=\...
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votes
1answer
15 views

How to find the basis of an extension field

Sorry for asking a simple question but why is it obvious that $\{1,\sqrt{3}+\sqrt{5}\}$ is a basis of $\mathbb{Q}(\sqrt{3}+\sqrt{5})$ over $\mathbb{Q}(\sqrt{15})$? I know that $[\mathbb{Q}(\sqrt{3}+\...
1
vote
1answer
20 views

Index of $G^n$ in $G$, an $s$-step nilpotent group of rank $\leq r$

For a group $G$ and $n \in \mathbb N$, let $G^n = \langle g^n \mid g \in G \rangle $ I am asked to show that if $G$ is $s$-step nilpotent and of rank at most $r$, then $[G:G^n] \leq n^{O_{r,s}(1)}$ ...
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0answers
5 views

Maximal separable subalgebras of semisimple associative algebras

Is anything known about maximal separable subalgebras of semisimple associative (and not necessarily commutative) algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic ...
0
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0answers
19 views

Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem?

Is it possible to derive Schur-Zassenhaus theorem (see e.g. https://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus_theorem) from Wedderburn-Malcev theorem (see e.g. https://www.math.uni-bielefeld.de/~...