Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

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Can two cycles be the inverse of each other if they are disjoint?

I am self-studying DF's Abstract Algebra and am currently working on the following exercise: Prove that the order of an element in $S_{n}$ equals the least common multiple of the lengths of the ...
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Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
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1answer
16 views

Conjugation of element lying in product of 3 groups lies in product of two groups.

I'm reading an article about tree automorphisms and I've got a problem whith something. Here it is: if $w \in \langle \gamma ^{h_1} \rangle \langle \gamma ^{h_2} \rangle \langle \gamma ^{h_1} ...
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28 views

Positively graded k-algebra

Suppose we have a positively graded $k$-algebra $A=\bigoplus_{i\ge 0}A_i$, such that $A_0$ has finite global dimension. Furthermore, all $A_i$ are finite dimensional and $A$ is generated in degree $0$ ...
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28 views

Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
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40 views

Examples of continuous non-transitive group actions

In studying topology, I encountered this problem: Let $S$ be a topological space and let $G$ be a topological group acting continuously on $S$ (group action as $G \times S \to S$ map is continuous). ...
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1answer
19 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order ...
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38 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
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37 views

The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
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1answer
40 views

2 question on solvable group's property

A theorem says a finite group G is solvable if and only if for every divisor n of $\vert G\vert$ such that (n,$\frac{\vert G\vert}{n})=1$,G has a subgroup of order n. Does this imply that G must ...
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35 views

how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
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39 views

On $O^{p'}(G)$ for a finite group $G$ (is my proof correct?)

Let $G$ be a finite group and let $N\unlhd G$. Consider $O^{p'}(G)$ which is the smallest normal subgroup of $G$ with factor group order coprime to $p$. Is $O^{p'}(G/N) = O^{p'}(G)/N$ if $N\le ...
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36 views

Best algebra text for Model Theory

I'm looking for an algebra book that is tailored towards some of the ideas in Model Theory, I'm currently slogging through Hodges' Model Theory. I'm a bit rusty with my algebra and was curious if ...
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1answer
39 views

When is $R^n / ( I \cdot R^n) = (R/I)^n$, for ring $R$ and ideal $I$?

I am looking at a set of old lecture notes in which I scribbled: $R^n / ( I \cdot R^n) = (R/I)^n$. However, I cannot recall why this is true. The setting is that $R$ is commutative, has identity, and ...
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17 views

Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
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90 views

$A\oplus C \cong B \oplus C$. Is $A \cong B$ when $C$ is finite, A and B infinite.

So my question is simply that for groups $A, B, C,$ if C is finite, A and B infinite and $A\oplus C \cong B \oplus C$, is $A \cong B$? My gut tells me this must be the case, and logically I can find ...
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1answer
57 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
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1answer
36 views

Minimal Polynomial of Algebraic Number

Suppose $\xi\in\mathbb{C}$ is an algebraic number, and suppose $m(x)\in\mathbb{Q}[x]$ is the minimal polynomial of $\xi$. If $\xi$ is a root of some monic polynomial $g(x)\in\mathbb{Z}[x]$, how can we ...
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34 views

Characterization of the transcendentals over a field

I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page: I couldn't solve any part of the equivalence, I think maybe ...
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1answer
51 views

Canonical embedding into dual space?

How would one go about proving that there is no embedding of a vector space into it's dual that is independent of a choice of basis? Thanks
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1answer
32 views

Does the relation $\pi(S_{i})=S^{-1}R-P_{i}\cdot S^{-1}R$ hold for prime ideals $P_i$ in a commutative ring $R$?

Let $R$ be a commutative ring. Let $P_{i}$, $1\leq i\leq n$ be prime ideals none of which are contained in each other. Let $S=R-(\cup_{i=1}^{n} P_{i})$. Then $S$ is a multiplicatively closed set and ...
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34 views

Question on quotient ring

Let $\zeta := \zeta_p = e^{2\pi i/p}\in \mathbb{C}$ and set $R = \mathbb{Z}[\zeta] = \left\{\sum_{i=0}^{p-1} a_i\zeta^i\mid a_i \in \mathbb{Z}\right\}$. Let $\mathfrak{p} = (1-\zeta)$ be an ideal of ...
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Help with Vector Definitions

How can I write a VERY brief statement that explains that you can redefine a vector space as a set V that is an abelian group under + and has scalars and a scalar multiplication that follow five ...
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66 views

The lattice points of a f.g. rational cone form a f.g. monoid.

In their book "Polytopes, Rings, and K-Theory" Bruns and Gubeladze sketch an alternative approach to Gordan's Lemma, which is stated in the headline (f.g. = finitely generated). I don't understand the ...
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4answers
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Does anyone know of a non-trivial algebraic structure satisfying these four identities?

Does anyone know of a non-trivial (i.e. cardinality $\geq 2)$ algebraic structure $(X,+,-)$ satisfying the following identities? $(x+a)-a=x$ $(x-a)+a=x$ $(x+y)+a = (x+a)+(y+a)$ $(x-y)+a = ...
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1answer
34 views

In general algebra, is every generating set equipotent to a finite basis itself a basis?

Question. Let $T$ denote an algebraic theory, and suppose $X$ is the $T$-algebra freely generated by a finite set $F \subseteq X$. Suppose $G \subseteq X$ also generates $X$ and that $|G|=|F|$. ...
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1answer
45 views

real closure of an archimedean field

my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!
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27 views

Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
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1answer
49 views

Which one of $\sqrt{3},\sqrt{-3},\sqrt 5,\sqrt{-5},\sqrt{15},\sqrt{-15}$ is the element of $\mathbb Q(\zeta)$

Let $\zeta = e^{2\pi i/15}$. Which one of $\sqrt{3},\sqrt{-3},\sqrt 5,\sqrt{-5},\sqrt{15},\sqrt{-15}$ is the element of $\mathbb Q(\zeta)$ Explain your answer. I get that using $e^{2\pi i/3}$ and ...
2
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1answer
55 views

what are some binary operations that commute?

Two binary operations $(\otimes, \oplus)$ commute if(?): $$ (a \otimes b) \oplus (c \otimes d) = (a \oplus c) \otimes (b \oplus d) $$ Firstly, is this the standard way of defining commutative ...
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53 views

Generalizing a statement about direct limits in the category of $A$-modules to other categories

The original problem is from Atiyah and Macdonald's Introduction to Commutative Algebra, Ex 2.15: Suppose $(M_\alpha,\mu_{\beta\gamma})$ is a direct system of modules over a unital commutative ...
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1answer
42 views

$a,b$ integral $\implies$ $a+b$ integral

I'm sure it's just silly thing. I'm reading Fulton's algebraic curves book and I don't understand this phrase of this proof: I didn't understand why according to the proposition we have $a\pm b,ab$ ...
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1answer
30 views

General lists of techniques to prove whether a set is a generator of a matrix group

It seems like a rather common problem in group theory, at least in undergraduate maths, to check whether a set is a generator of a group. The question is usually as follow: Given a group $G$, and a ...
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1answer
68 views

What are the non-trivial normal subgroups of $O(3)$?

What are the non-trivial normal subgroups of $O(3)$? My guess is that the only one is $SO(3)$, but it's really only a guess, based on the fact that $O(3)$ is disconnected 3-manifold of $SO(3)$ and the ...
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1answer
64 views

Structure of a group, $G$, of order $pq$ where $p, q$ are prime.

There is a proposition in Beachy and Blair's Abstract Algebra that I don't entirely follow. The proposition is the following: Let $G$ be a group of order $pq$, where $p > q$ are primes. a) If ...
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3answers
32 views

Ideals of the residual classes $\mathbb Z_n$

Let $n$ be a positive integer and considere the ring $\mathbb Z_n$ of the residual classes modulo $n$. My intuition tells me that there is no two distinct ideals of $\mathbb Z_n$ with the same number ...
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32 views

Characterising subgroup

Let $\omega $ be a path in $\hat{X}$ with $\omega(0), \omega(1) \in p^{-1}(x_0)$, where $p$ is a covering map $p:\hat{X} \rightarrow X$. Let $\alpha=[p \circ \omega] \in \pi_1(X,x_0)$. Then we have ...
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21 views

Finding this Polynomial Subspace

Let $A = k[x^{\pm 1}, y^{\pm 1} ] $, considered as a $k$ - algebra. Can someone give me a nice description of the (vector) subspace: $$ A_0 = \lbrace (f,g) \in A^2 : \frac{ \partial f}{\partial y} = ...
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1answer
79 views

Cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ as a continuous group or a Lie group?

Is there any example of Borel cohomology group $H^n(G, \mathbb{R}/\mathbb{Z})$ for any $G$ such that $H^n(G, \mathbb{R}/\mathbb{Z})$ is a continuous group? Such as a Lie group? Most of the examples ...
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co and contravariant vectors, their difference and properties

Very often when talking about covectors, co- and contravariant stuff, it's mentioned that there is no difference in "normal" linear algebra. That the difference only comes "when dealing with curved ...
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Proving that a field $K$ can be generated by algebraically independent elements and an separable element

Let $k$ be a perfect field (either $k$ has characteristic $0$, or characteristic $p > 0$ and every element has a $p$th root), and let $K$ be a finitely generated extension field. I have a question ...
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1answer
32 views

Is a torsion-free discrete abelian group of finite rank isomorphic to a subgroup of $\mathbb{Q}^k_d$?

If $G$ is a torsion-free discrete abelian group of rank $k$, then is it true that $G$ is isomorphic to a group $H$, where $\mathbb{Z}^k < H < \mathbb{Q}^k_d$? Here, $\mathbb{Q}^k_d$ is the ...
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1answer
23 views

Irreducibility in Galois/non Galois Extensions

Let $k$ be a field and $\alpha$ algebraic over $k$. Let $K$ be the Galois closure of $k(\alpha)$ (obtained by adding all conjugates of $\alpha$). If $f(x) \in k[x]$ is irreducible over $k[\alpha]$ is ...
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1answer
60 views

Projective special linear groups

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
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In an Integral Domain, every prime is an irreducible. Flaw in the Proof?

In an Integral Domain, every prime is an irreducible. The proof is as follows : Let $D$ be the integral domain, then, if $a \in D$, it's possible to express a = $bc$ where $b,c \in D ...(1)$. ...
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31 views

When a system of rational linear homogeneous equations have complex solutions

Question: When a finite system of rational linear homogeneous equations in finitely many variables have a nontrivial complex solution (that is not a rational solution), does it imply that there is ...
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1answer
45 views

Additive inverse in a field is unique

there will be an $a$ let's prove $-a$ is unique. let's assume there are 2 additive inverses $b$ and $c$ therefore $a+b=a+c$ let's multiply them by the multiplicative inverse of $a$ I try to use the ...
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1answer
19 views

Let x, y be integers. Show that if x = y (mod n), then x + mZ = y+mZ,and conversely, if x+mZ=y+mZ then x = y (mod n)?

Let $x, y$ be integers. Show that if $x = y\mod n$, then $x + nZ = y+nZ,$ and conversely, if $x+nZ=y+nZ$ then $x = y\mod n$? I have no a clue on how to prove this! Please help.
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1answer
46 views

Show that the set $U_{77,76}$ of solutions to $x^{76} = 1$ is a subgroup of $U_{77}$.

Show that the set $U_{77,76}$ of solutions to $x^{76} = 1$ is a subgroup of $U_{77}$. I don't understand this question at all but could someone also explain what it means by $U_{77}(76)$.
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1answer
40 views

A Fixed Field of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I was working on a review problem from Dummit and Foote and came across the following issue. It is clear that the Galois group of the splitting field for the polynomial $(x^2-2)(x^2-3)(x^2-5)$ has ...