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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Elementary geometry from a higher perspective

I'm searching for some references that deal with topics from "elementary geometry" analysing them from a "higher" perspective (for example, abstract algebra, linear algebra, and so on).
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Abstract Algebra Note-taking questions

I have a question about how to take notes. Should I copy down every theorem, or can I only copy down important ones and refer to the book for the rest? Same goes for proofs. Should I try to memorize ...
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GCD's and how they generate groups

I was reading something today an it was talking about $U_{15}$, all the integers relatively prime to 15, and how it was generated by the set {7,11}. I understood it all, but I thought that if the ...
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Example of a Schauder basis for $C^0(\mathbb{R})$

Can someone please provide me with an example of a Schauder basis for $C^0(\mathbb{R})$. If there isn't one could you please explain why not. My understanding of basis for infinite dimensional vector ...
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When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
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Why $( Z_3\rtimes Z_2)\times Z_2 \cong (Z_3\times Z_2)\rtimes Z_2$?

I got an explanation, it says as $Z_2$ is in the kernel of the homomorphism. But I can't understand from that. Also can you tell me why $Z_3\rtimes Z_2\cong S_3$ ? Thank you.
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Proof of an isomorphism

I'm trying to prove that if $A=\mathbb{Z}_6$ and $S_1=\{\bar{1},\bar{2},\bar{4}\}\subset\mathbb{Z}_6$ then $AS^{-1}\simeq \mathbb{Z}_3$. What can I do to prove it?. My idea consists in proving that ...
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Atiyah-Macdonald 5.2

Exercise 5.2 in Atiyah-Macdonald asks to show the following: "Let $A$ be a subring of a ring $B$ such that $B$ is integral over $A$, and let $f: A \to \Omega$ be a homomorphism of $A$ into an ...
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Irreducibility in Rings [on hold]

$f(x,y)=x^{2}-y^{2}+1$. Is it irreducible in $\mathbb{C}$? In $\mathbb{Z}_{7}$? Please include the reason and the result. I have read Eisenstein Criterion and many other theorems. None of them ...
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63 views

Does a Group being Finite Imply that It Is Cyclic?

I have been studying Abstract Algebra, and all the finite groups that we have studied so far have also been cyclic. So, is it true that all finite groups are cyclic? If yes, what is the theorem? If ...
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Defining an isomorphism

If we have to prove that the multiplicative group of integers modulo $8$, $U(8)$, is isomorphic to a set of matrices, are we allowed to define the isomorphism by saying: $$\begin{align} ...
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Is $\Bbb{R}[X,Y]/(X^2+Y^2)$ a UFD or Noetherian?

Hello everyone I would like to know if $R$:= $\Bbb{R}[X,Y]/(X^2+Y^2)$ is UFD or Noetherian. I'm not really confortable in seeing how $\Bbb{R}[X,Y]/(X^2+Y^2)$ looks like. From what i've ...
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18 views

Introduction to Reidemeister--Schreier Method

I am learning Reidemeister--Schreier method, a method determining explicitly presentations for subgroups of a given group. Can anyone recommend some introductory material, preferably those with ...
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13 views

polynomial algebra elements [on hold]

Let we have the polynomial algebra $A(n,m)$ for $n \in\mathbb{N}$ and $m\in \mathbb{N}^ {n}$. Let F be a field of characteristic p (p is not zero). For $a\in \mathbb{N}^ {n}$ we have $X^a for ...
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Prove that T is not a zerodivisor in A[T]

Let A be any ring, consider the polynomial ring A[T]. Prove that T is not a zerodivisor in A[T]. Generalise the argument to prove that a monic polynomial $$ f=T^n+a_{n-1}T^{n-1}+\dots+a_0 $$ is ...
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23 views

To make a polynomial with coefficients in a finite field uniform at random

We define the polynomial ring $R[x]$ consist of all polynomial with coefficients from $\mathbb Z_p$. Let $P_1$ be a polynomial such that $P_1 \in R[x]$. The aim is to make $P_1$ uniformly at random. ...
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greatest common divisor in the ring $F(x)[y]$

Assume $F$ is a field (we can assume here $F = \mathbb R$), and let $R = F(x)[y]$. If we now know that two elements $a,b$ in $R$ have no common divisor other than $1$ we can write $\gcd(a,b) = 1$. ...
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Cancellation laws in Rings

In rings left and right cancellation laws generally don't hold. can anyone generalize some cases so that we are ensured when the cancellation laws hold in rings?(the case I found was in Integral ...
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quotient $\frac{Z[i]}{(3-i)}$ is isomorphic to??? [on hold]

if $Z[i]$ is the ring of gussian integers, the quotient $\frac{Z[i]}{(3-i)}$ is isomorphic to which ofthe following 1.$Z$ 2.$\frac{Z}{3Z}$ 3.$\frac{Z}{10Z}$ 4.$\frac{Z}{4Z}$ How to solve this?
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Tensor Product and Splitting

Let $M$ be a $\mathbb{Z}$-module and consider the submodule $K=\langle m\otimes m\mid m\in M\rangle$ of $M\otimes M$. Under what conditions does the SES $0\to K\to M\otimes M\to M\wedge M\to 0$ split? ...
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The Diagonal Subgroup of $A \times A$ is Maximal iff $A$ is Simple

Let $A$ be a group and $G = A \times A$. Define $D= \{(a,a,)\mid a \in A\}$ (the diagonal subgroup of $G$). Prove that $D$ is a maximal subgroup of $G$ if and only if $A$ is simple, i.e. it has no ...
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A question about tensor product over rings .

Let $A,B,C$ be three rings such that $f:A\to B$ and $g:A\to C$ are ring homomorphisms. How is $B\otimes_A C$ defined? I am especially worried about how $b\otimes_A tc$ is defined, where $t$ is a ...
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Characteristic of ring

What are characteristic of following ring? $R$=$M_2[Z]$ $R$=$Z_4$$\times$$4Z$ Is the characteristic of (1) is zero and (2) is (4,0)?
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Find the number of prime ideals?(CSIR 2014)

Let $p,q$ be distinct primes. Then (1) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct ideals. (2) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct prime ideals. (3) $\dfrac{\mathbb{Z}}{p^2q}$ ...
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Submodule iff subgroup?

It is late at night and time for another silly question: Is it true that a subset $S$ of an $R$-module $M$ is a submodule if and only if it is a subgroup of $M$ as an abelian group? Of course, by ...
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Bounding the order of a group by its nilpotentizer

Let $G$ be a finite non-nilpotent group. We put $nil_G(x)=\{y\in G\mid \langle x,y \rangle \text{ is nilpotent}\}$, called the nilpotentizer of $x$. Note that $nil_G(x)$ may not be a subgroup of $G$, ...
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Group homomorphisms between $D_4$ and $Z_2 * Z_2$, and between $D_4$ and $Z_4$?

Here $D_4$ is the group of symmetries of the square, with order 8. I'm not sure how to go about specifying homomorphisms between non-cyclic and cyclic groups.
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Fundamental Theorem of Finitely Generated Abelian Groups and $Z_4$

Under my understanding of this theorem, $Z_4 \cong Z_2*Z_2$, but this is obviously false. However, 2 is prime, so why does this not fall under this theorem?
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All homomorphisms between $Z_{12}$ and $Z_{18}$

I know homomorphisms between cyclic groups are determined by the mapping of the generator of the domain. Taking the generator 1, I know that since 12*1=0, I need to find multiples of 12 that are 0 mod ...
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Prove that if $f : \mathbb Z_p \to G$ is a homomorphism, then $f$ is either injective or trivial(i.e. $f(x)=1$ for all x).

I'm stuck on the last part that I assume there is one element other than $0$ and $1$ belongs to the kernel and I try to prove that $f(1)=1$, but I didn't see any clue to do that.
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Prove there is no surjective homomorphism between $D_4$ (the symmetries of the square) and $Z_4$

I've been trying to figure out this homework problem for a while now and always get stuck. Both groups have 2 elements of order 4, and the remaining non identity elements have order two. Since the ...
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Prove $(aba)^k$ = $ab^ka$ for all K in W iff $a^2 = e$.

Prove $(aba)^k$ = $ab^ka$ for all K in W iff $a^2 = e$. a,b are elements of a group. I'm not sure where I am supposed to start.
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Covering Spaces in Representation Theory.

I'm reading the paper "Covering Spaces in Representation Theory" of K. Bogartz and P. Gabriel. Now I'm in section 2, proposition 2.3, on the first three lines concludes that the functor $l \mapsto ...
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Prove if 0(a) is odd, show a is a square.

Let $G$ be a group and a is an element of $G$. If $o(a)$ is odd, show $a$ is a square. I started by supposing that $o(a) = 2n+1$ which implies $a^{2n+1}=1$. I am not sure if I am on the right track ...
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Ways to find the order of an element in a group

Is there a better way of finding the order of an element in a group other than circling until the identity is reached? Is there or CAN there be a better general ways of finding orders of elements? ...
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Is it true, $O(ab)=O(ba),$ Where $G$ is a group and $a,b \in G.$

Suppose $O(a)$ and $O(b)$ is finite and also $O(ab)$ and $O(ba)$ is finite. Then L.C.M $(|a|,|b|)= L.C.M (|b|,|a|).$ (Is that Correct ?) Suppose $O(a)$ and $O(b)$ is finite but $O(ab)$ is infinite, ...
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Showing that the $\text{ord}(a+b) = \text{min}(\text{ord}(a),\text{ord}(b))$ in a DVR

This is Problem 2.29 from Fulton's Algebraic Curves. First a bit of background because I don't know how standard his terminology is. For a discrete valuation ring $R$ with maximal ideal ...
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If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$

If $G_1\cong G_2$ and $H_1\cong H_2$ then $G_1 \times H_1 \cong G_2 \times H_2$ Proof: $f_G:G_1\rightarrow G_2$ and $f_H:H_1\rightarrow H_2$. Question 1: Is the following statement valid? Does ...
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Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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Proof, that $(\mathbb R×\mathbb R;+,*) $ is a ring, but not a field

How do I prove, that $(\mathbb R×\mathbb R;+,*) $ is a ring, but not a field, where the $+$ and $*$ operations are: $(a,b)+(c,d):=(a+c,b+d)$ and $(a,b)*(c,d):=(ac,bd)$? For the solution: so I would ...
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For a general group what does the formula $\sum_{g\in G} |G|/ord(g)$ mean? [on hold]

In my research, I have come across this group formula, $\displaystyle \sum_{g\in G} \frac{|G|}{ord(g)}$. Has anyone seen this before? Where have you seen this formula before? I am wondering if ...
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If $G,H$ are groups then $G\times H\cong H\times G$

This seems like a basic question, but I searched for a while and couldn't find it on the site. I want to know if I have a valid proof for the following theorem. If it is correct, I'd like to see how ...
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Express $(a*b)^{-1}$ in terms of $a^{-1}$ and $b^{-1}$.

Express $(a*b)^{-1}$ in terms of $a^{-1}$ and $b^{-1}$. a and b are elements of a group. * is the operation. I know $(a*b)^{-1}$ is not equal to $a^{-1}$ * $b^{-1}$. I also understand that ...
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Solving $-\alpha x^2-\beta y^2-\alpha\beta z^2=-p$ to check for isomorphism of quaternion algebras.

Edited I am considering quaternion algebras $(-\alpha,-\beta)_\mathbb Q$, with $\alpha,\beta>0$. I am trying to find several of these, which are not isomorphic. I do not want to digress into ...
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Finding an isomorphism from $\mathbb{R}^\times$ to a defined group $G$

Here's the problem I am solving: $G=\{x\in \mathbb{R}:x\not = 0\}$. The operation for $G$ is "$*$", with $x*y=\frac{1}{2}xy.\mathbb{R}^\times$ is the multiplicative group $\mathbb{R}.$ Find an ...
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Maximal normal subgroup has prime index

I am trying to solve the following exercise taken from Rotman's An Introduction to the Theory of Groups: Let $M$ be a maximal subgroup of $G$. Prove that if $M \lhd G$, then $[G:M]$ is finite and ...
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The order of a is the same as the order of a−1 [on hold]

If a,b,c belong to G than show The order of a is the same as the order of a−1.
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Is this statement about simple finite group true?

I have proven it by myself, so I'm not sure whether I proved it right or wrong. That is: Let $G$ be a simple finite group. Let $H$ be a subgroup of $G$ such that $H\neq G$. Then, $G$ is ...
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30 views

Meaning of a commuting maps?

What is commuting and commuting maps in mathematics? Did they different with commutative group?
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Are smooth varieties locally isomorphic to the affine space?

A smooth $n$-dimensional manifold is locally isomorphic to $\mathbb{R}^n$. I am wondering if the analogous statement for smooth algebraic varieties is also true. Let $X$ be an $n$-dimensional ...