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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Introducing multiplication of cosets

So, i have encountered two ways to introduce the multiplication of cosets, and i want to understand exactly what is happening in each, specifically in light of the multiplication of cosets being ...
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What the difference between $A/m$ and $A_0$

$A$ is a integral domain. $m$ is a maximal ideal . $A_0$ is the localization of $A$ by $A-0$.(Field of fractions) $A/m$ is the quotient at $m$. What the difference between $A/m$ and $A_0$?
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General question about quotient rings

I hope to better understand the notion of a quotient ring through this example: I am given $R=\mathbb{Z}[i]=\{a+bi:a,b\in \mathbb{Z}\}$ and $M=\{a+bi: 3|a,3|b\}$. I have already shown that $M$ is a ...
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In S_3, an even permutation must have an odd number of orbits. True or False?

Well, somehow i considered the unit cycle of S_3 : (123). And since it can be expressed as a product of 2 transpositions mainly: (12)(13) which i believe proved S_3 to be an even permutation. I have a ...
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77 views

Proof of Wedderburn's Theorem

I've been going through a proof of Wedderburn's theorem: and I'm stuck on the very last part, where the author refers to example 2.1.4. (linked below). I don't understand what $D^n$ means, or why it ...
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How do i prove that $\gcd(s,n)=\gcd(t,n) \Rightarrow <a^s>=<a^t>$?

Let $G$ be a finite cyclic group generated by $x$. $(|G|=k)$ Let $n,m\in\mathbb{Z}$ such that $\gcd(n,k)=\gcd(m,k)$. Then, $<x^n>=<x^m>$. I can prove the converse, but i ...
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Counterexample to “A/I is Artinian, when I is the annihilator of Artinian A-module”.

Let M be an Artinian A-module and let I be the annihilator of M in A. Is A/I necessarily an Artinian ring? I believe the answer is no since this comes off of a similar result regarding Noetherian ...
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26 views

Show that if $N$ is a normal subgroup of $G$ which contains all commuters then $G/N$ is abelian.

I am working on my proof for class and I was wondering if this look ok? Let $N$ be a normal subgroup of $G$ we want to show that $G/N$ is abelian, or $(aN)(aN) = abN = baN = (bN)(aN)$. Since $N$ ...
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44 views

Can we replace the $B$ to $A$ in this proposition

I am working through Atiyah's Commutative algebra and am having question with the following proposition: $\text{Page 63:}$ Proposition 5.15. Let $A$ $\subseteq$ $B$ be integral domains, $A$ ...
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57 views

Generators and relations as a functor [closed]

Make “generators and relations” into a functor. What is its left adjoint? [Bergman] How could one do this?
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14 views

Mono unary algebra and their product

Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$. This describes a mono unary algebra. Now I wish to see the structure of $A_3\times A_5$. Clearly ...
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27 views

Exhibiting infinitely many subfields of the extension $\mathbb{F}_{p}(x,y) / \mathbb{F}_{p}(x^{p},y^{p})$ with this method.

Suppose that we want to show that $\mathbb{F}_{p}(x,y) / \mathbb{F}_{p}(x^{p},y^{p})$ is not a simple extension by showing that there are infinitely many intermediate subfields. I recently posted a ...
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73 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...
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36 views

Is this statement true? (Cyclic Group)

Let $G$ be a cyclic group. Let $a$ be an element of $G$ such that $a\neq e$. If there exists $m\in\mathbb{Z}^+$ such that $a^m=e$, then $G$ is finite. Is this true? Moreover, is ...
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32 views

Proving that if $n\times n$ Hadamard matrix exists, then 4 divides $n$

Im looking for an explanation of the following: a standard way to prove that, if there exists Hadamard matrix of dimension $n > 2$, then $4|n$, is to suppose that without loss of generality every ...
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57 views

Surjective homomorphism on Commutative Ring

Let $A$ be a commutative ring, $R= A[x_{1},...,x_{n}]$ and $(a_{1},...a_{n}) \in A^{n}$ . Let $\phi : R \to A$ be defined by $\phi (f(x_{1},...,x_{n}) = f(a_{1},...,a_{n})$. Then show that $\phi$ is a ...
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30 views

How to show Not a Free Module

Let $\mathbb K$ be a field, $A= \mathbb K [x,y]$ and $ M = Ax + Ay$. prove that $M$ is NOT a free module!
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How is this subgroup normal?

Let $G$ be a group, and let $U$ be a subset of $G$. Let $\hat{U}$ be the smallest subgroup of $G$ containing $U$. Then $\hat{U}$ is the intersection of the collection of all the subgroups of $G$ ...
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17 views

Extensions of PID [duplicate]

Let $R$ be a principle ideal domain. One can consider it's localization $F\supset R$ which is a field so it is a principle ideal domain too. I wonder if every ring $A$ such that $R\subset A\subset F$ ...
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40 views

“Self invertible” group

Let there be an Abelian group with a binary operation $\ast$ on a set $S$. Let such a group respect the following propriety: $$ (X\ast Y)\ast Y = X$$ For any $X$ and $Y$ in $S$. I realize that by ...
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51 views

Show module is Noetherian

Let $R$ be a commutative ring and let $0 \to L \to M \to N \to0$ be an exact sequence of $R$-modules. Prove that if $L$ and $N$ are noetherian, then $M$ is noetherian. I tried considering the pre ...
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My professor says that this equation in a finite field has a solution but I don't think it does.

More than likely it is I who is mistaken, but is there a chance that my professor made a mistake in the following problem? We are tasked with: Let $p = 3$. We do not have an element of order $5$ in ...
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26 views

what does this triangle-like notation mean?

$$\triangleright\quad \text{and}\quad \triangleleft$$ I saw these notations in some abstract algebra texts and i don't know what does this mean. What do they mean?
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Determinant of block matrices

A couple of questions regarding the solution Why is it that we are able to prove this just by induction on s and not t simultaneously? For the green underline, why are we able to use the standard ...
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18 views

Definition of “Cayley table”

Dummit-foote p.21 Let $G=\{g_1,\cdots,g_n\}$ be a finite group with $g_1=1$. The Cayley table of $G$ is the $n\times n$ matrix whose (i,j)-entry is $g_ig_j$. This definition is based on an ...
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Direct proof for the independence of $\operatorname{Tor}$

It is known that $\operatorname{Tor}$ is independent of the choice of the resolution. More specifically, I am trying to do the exercise 1 (c) of Vick's homology theory. The author gives the ...
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30 views

What exactly are the elements of $\mathbb{Z}_p[x]/\langle p(x) \rangle$?

It is wellknown that for a polynomial ring $\mathbb{Z}_p[x]$, $\mathbb{Z}_p[x]/\langle p(x) \rangle$ for prime $p$ is a field if and only if $p(x)$ is irreducible over the given polynomial ring, in ...
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31 views

Surjective Homomorphism Symmetric group

For $G=S_4$ i'm having a bit of trouble following the solution. For the blue underline I was wondering if there is a strategy for spotting this relatively quickly. For the green underline I ...
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53 views

Basis for $\mathbb{Q}[\sqrt{8}]$ over $\mathbb{Q}[\sqrt{2}]$

Provided that $x^2-8$ is the minimal polynomial for $\mathbb Q[\sqrt8]$ and $x^2-2$ is minimal for $\mathbb Q[\sqrt 2]$ we should have a basis with four elements. Thus far I know $1$ and $\sqrt 2$ ...
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19 views

Divisible Direct Sum or Direct Product

We know that direct sum and direct product of divisible R-modules are divisible when R is a domain. Does there exist non-divisible modules with their product or sum being divisible? Or, if R is not a ...
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25 views

Property of a quotient of a nonfaithful module over a Dedekind ring.

I'm confused about the following fact. Suppose $R$ is a Dedekind domain, and $M$ a module which is not faithful, i.e., there exists $a\neq 0$ such that $aM=0$. Then if $D$ is the set of elements not ...
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If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic

(i) If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic (ii) Prove that $U(p^n) \thickapprox Z_{(p^n -p^{n-1}) }$ Solution : If I prove that if $U(p^k) $ is cyclic, then, we know that ...
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15 views

Order of a Group Help Abstract [duplicate]

Is the order of the Heisenberg group infinite since H = 1 a b 0 1 c 0 0 1 under matrix multiplication where a,b,c are real numbers? How would I formally state this?
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Abstract Algebra Order of Group Help

I am doing a group writing project on the Heisenberg Group and one of the things we have to find if the order of the group is finite or infinite. How do I begin to find the order of a group? I am ...
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(12345) is an even permutation of S_5. True or False?

The answer i had for this question was True, yet i'm not sure. Well, from what I know so far was that: $(12345)$ can be expressed as a number of 4 transpositions such as: $(12)(23)(34)(45)$ which is ...
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35 views

What is the meaning of an algebra?

An algebra $A(*,\hat{} ,\sim)$ is said to be Boolean algebra if it satisfies some conditions...In this statement what is the meaning of starting word an algebra?
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33 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
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1answer
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Is $x^4+2$ irreducible over $\Bbb{Q}(i)$?

Let $f(x)=x^4+2$. Using the Eisenstein test to $f(x+2)$, one can show that $f$ is irreducible over ${\Bbb Q}$. Let $\beta$ be a complex root of $f$. Then the question in the title is equivalent to ...
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30 views

$p$-subgroups conjugate iff $\cong$ to Sylow p-subgroups of some other groups?

Let $G$ be a finite group and $p$ a prime such that $p^\alpha$ divides $|G|$ and $p^{\alpha+1} \nmid |G|$. I know that Sylow $p$-subgroups of $G$ are conjugate to one another but if we have some ...
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1answer
23 views

order of an element formula

I was wondering whether there's a formula or something. If it is given that $x^n = e$ and $x^m = e$, does it mean $x^{gcd(n,m)} = e$, so we can determine whether $x=e$ or $x \ne e$?
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Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
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Conjugate subgroups of $S_4$

$A = \langle (1,2,3),(1,2)\rangle$ $B = \langle (1,2,4),(1,2)\rangle$ $C = \langle (1,3,4),(1,3)\rangle$ $D = \langle (2,3,4),(2,3)\rangle$ I want to proof that these subgroups of $S_4$ ( which ...
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proof a function is an isomorphism

When we prove a function is an isomorphism, we need to prove it's a bijection and it's closed under an operation. In one example I had no problem proving the first part, but in the second part, I ...
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About the definition of the tensor product of modules

I was reading something about tensor product (balanced product) of modules (over an arbitrary ring $R$), but I cannot realize why we need a left and a right module. Would it be the same with two right ...
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Dummit and Foote page 526

I'm having trouble with a line of example 2 on page 526. Consider the field $\mathbb{Q}(\sqrt{2},\sqrt{3})$. generated over $\mathbb{Q}$ by $\sqrt{2}$ and $\sqrt{3}$. Since $\sqrt{3}$ is of degree ...
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How do I show a mapping is a homomorphism?

I don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and more ...
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order of elements in direct product

I have a conceptual question: if a group has 1 element of order 1 and 1 element of order 2 (e.g., nonzero reals), what changes if your take its direct/cartesian product as a group?
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Is the Axiom of Choice implicitly used when defining a binary operation on a quotient object?

Let's say you have a group $(G,\cdot)$ and you have a normal subgroup $N$ (note we are considering this only as a set). And now we want to define a binary operation $\star$ on $G/N$ such that $(G/N, ...
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kähler differentials of power series ring

Let $K$ be a $p$-adic field (for example a finite extension of $\mathbb{Q}_p$) and let $A = \mathcal{O}_K[[X_1,\ldots,X_n]]$ which we consider as a topological ring. I would have thought that the ...
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Checking the maximality of an ideal

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...