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

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Which of the following about a permutation is correct?? (CSIR-2015, June)

Let $\sigma:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ be a permutation (one-to-one and onto function) such that $$ \sigma^{-1}(j)\le \sigma(j) \quad\text{for all $j$ such that }1\le j\le 5. $$ Then ...
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0answers
11 views

Normal Submagma?

Is there a definition of normal submagma? visit https://en.wikipedia.org/wiki/Magma_(algebra) For normal sub-quasi-group I found two: A sub-quasi-group $H$ is called normal if there exists a normal ...
2
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0answers
26 views

Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
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1answer
11 views

Explicitly compute the trace for the tautological representation of $D_4$ of $\mathbb{R}^2$.

Fix a finite dimensional representation $\rho: G \longrightarrow GL(V)$ of $G$. Its trace is defined as the function $tr:G \longrightarrow F$ defined by $tr(g) = tr(\rho(g))$. Explicitly compute ...
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2answers
62 views

Hardcore Abstract Algebra Book Request

I am going to start learning Abstract Algebra soon. I was originally going to start with Dummit and Foote, but I am starting to abandon that idea. I want to use a "hardcore" algebra book. I don't mind ...
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1answer
21 views

Non-zero maps between modules

Let $R$ be a ring and $M,N$ two $R$-modules such that there exists a non-zero map $\psi : M \to N$. Is it true that there exists a non-zero map $\varphi: N \to M$ ? I do not have any particular ...
4
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0answers
27 views

The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
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1answer
42 views

Follow up question to finding primes $p$ such that $f(x)=x^6 - x^3 +1$ factors (in various ways) in $\mathbb{F}_p$

I asked this question yesterday, however, I am not sure how to compare the solution given on this site to the "worked example" solution as in my notes. $\textbf{Problem Statement:} $ Let $f(x)= x^6 - ...
5
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2answers
62 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
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0answers
24 views

Galois group isomorphic to $\mathfrak{S}_5$.

Let $f(x) \in \mathbb{Q}[x]$ an irreducible polynomial of degree $5$ and with splitting field $K \supset \mathbb{Q}$. If $\mathbb{Q}(\sqrt{7})$ and $\mathbb{Q}(\sqrt{11})$ are subfields of $K$,is ...
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0answers
33 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that ...
4
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4answers
71 views

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5

what is the greatest integer that divides $p^4-1$ for every prime number p greater than 5(this is a gre subject math problem) I think that $p^4-1=(p^2+1)(p-1)(p+1)$,so 8 must divide all the $p^4-1$ ...
2
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1answer
35 views

Subgroup proof verification.

Let $G$ be an abelian group, K is a fixed positive integer. $H$={$a\in$ $G$ $|$ $|a|$ divides K} . Prove that $H$ is a subgroup of $G$. My way of proving (Let me know how I could make it better or ...
3
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1answer
51 views

Exercise from Serre's “Trees” - prove that a given group is trivial

In Serre's book "Trees" on page 10 the following exercise is given: Show that the group defined by the presentation $$x_2x_1x_2^{-1}=x_1^2, \hspace{7pt} x_3x_2x_3^{-1}=x_2^2, \hspace{7pt} ...
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0answers
38 views

Let I be an ideal of a ring R. Prove that the quotient ring R/I is a commutative ring if and only if ab − ba ∈ I for all a, b ∈ R. [on hold]

This is the report no. 3 of Jennylou Canlas in our subject math126 in MSU Proof: Suppose R/I is a commutative ring. Let a, b ∈ R. Then (a + I), (b + I) ∈ R/I. Since R/I is commutative , (a + I)(b ...
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2answers
23 views

Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. [duplicate]

I'd like a proof of: Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. This is my report no. 3 in my subject ...
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1answer
54 views

Are $\mathbb{N}$ is isomorphic to $\mathbb{Q}$? [duplicate]

Are $\mathbb{N}$ isomorphic to $\mathbb{Q}$? There are any difference between isomorphism and cardinal equality? If $X$ and $Y$ are two sets and $\text{Cardinal}(X)=\text{Cardinal}(Y)$, is $X$ ...
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1answer
36 views

Question about the quotient map of commutative algebra [on hold]

Let $A$ be a commutative algebra and let $I$ be an ideal of $A$. Denote the quotient map from $A$ to $A/I$ by $f$. Let $J$ be a maximal ideal of $A/I$. Is it true that $f^{-1}(J)$ is a maximal ideal ...
6
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1answer
39 views

Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
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1answer
79 views

Group as a $\mathbb Q$-vector space

Let $G$ be a torsion free abelian group of having $n$ number of maximally rationally independent elements $r_{1}, r_{2}, ..., r_{n}$ and assume that $G$ is not finitely generated. Is this correct ...
1
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1answer
32 views

Prove that there's a unique morphism that completes the commutative diagram

I have to prove that there's a unique $\gamma : M'' \rightarrow N''$ that completes this diagram considering the rows are exact. $$\begin{array} MM' \stackrel{f_1}{\longrightarrow} & M & ...
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1answer
54 views

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$

Prove that $\mathbb{Q}{[\sqrt3]} = \lbrace a + b\sqrt3: a, b \in \mathbb{Q} \rbrace$ is a subfield of $\mathbb{R}$ My answer: Must show: i) $0 \in \mathbb{Q}{[\sqrt3]}$ ii) $1 \in ...
0
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1answer
24 views

Are there infinite-dimensional, artinian C*-algebras?

A ring is artinian if it has no infinite descending chains of ideals. Of course finite-dimensional algebras are artinian. I'm wondering if it's possible to have an artinian C*-algebra (or Banach ...
4
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3answers
44 views

Find a prime $p$ such that $f(x)=x^6 - x^3 +1$ factors in to linear factors in $\mathbb{F}_p[x]$

Find a prime $p$ such that $f(x)=x^6 - x^3 +1$ factors in to linear factors in $\mathbb{F}_p[x]$ $\textbf{My attempt:}$ Notice that $f(x)$ is the $18$-th cyclotomic polynomial, $\Phi_{18}(x)$. For ...
2
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1answer
42 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
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1answer
19 views

Another approach to $A/I\otimes_A A/J\simeq A/(I+J)$?

The same question appears here $A/ I \otimes_A A/J \cong A/(I+J)$ however I'm looking for a different approach. Let $A$ be an algebra, $I$ a left ideal and $J$ a right ideal. I'd like to show ...
2
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1answer
49 views

Looking for various proofs there is no faithful representation $\rho:S_4\rightarrow GL(\mathbb{R}^2)$

I stumbled on the question when thinking about a representation $\rho:D_4\rightarrow GL(\mathbb{R}^2)$ as symmetries of the four points $\{(\pm 1, \pm 1)\}$. There didn't seem to be a good geometric ...
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3answers
66 views

Find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $.

I'd like to find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $. What I've tried so far: I tried to do $ \varphi(Id) = \bar{0} = \bar{4} $ (as somebody used here). But then I realized ...
2
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2answers
37 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
4
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2answers
45 views

Groups of the from $gMg$ in a monoid where $g$ is an idempotent

Let $(M, \cdot)$ be a finite monoid with identity $e$. It is easy to see that $gMg = \{ gxg : x \in M \}$ forms a monoid with identity $geg = g$ if $g$ is an idempotent. If $gMg$ contains no ...
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1answer
58 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
7
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1answer
84 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
0
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1answer
33 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
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1answer
12 views

dimension of the quotient of a bialgebra

I am stuck in a proof of a lemma that I am in need of. The situation is as follows: Let $k$ be a field and $A$ and $B$ two finite-dimensional $k$-bialgebras, where the dimension of $A$ is a prime ...
3
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2answers
42 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
2
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1answer
49 views

When is $M\otimes N$ a module?

So suppose we have a $R$-$S$-bimodule $M$, and an $S$-$T$ bimodule $N$, then we can construct the abelian group $M\otimes_S N$. Under what conditions could we make this a module? I would expect this ...
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1answer
14 views

Every modular right ideal is contained in a modular maximal ideal

If $R$ is a ring, possibly without $1$, a right ideal $\mathfrak{a}$ of $R$ is modular if there exists $e\in R$ such that $r-er\in \mathfrak{a}$ for all $r\in R$. So $e$ is a left identity mod ...
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8answers
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What's a group whose group of automorphisms is non-abelian?

I recently attended an interview for admission to graduate programs in Mathematics. The interviewing professor asked me a question - Tell me a group whose group of automorphisms is non-abelian. ...
3
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0answers
22 views

Finding binary operations on connected graphs

If $G = (V,E)$ is a connected graph with $||V|| \geq 2$ , $W(G)$ being the set of all paths in $G$. How do you find a binary operation $ +$ on $W(G)$ such that $\langle W(G),+\rangle$ is an algebra ...
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2answers
51 views

Proving associativity in Algebra

How to proof that a specially defined Transitive Join for the relations $R \subseteq A$ x $B$ und $S \subseteq B$ x $C$ is associative? The join is defined as: $R \Join S =_{def} \{(a,c)| $ there is ...
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0answers
28 views

Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
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1answer
16 views

Ring of polynomials as free module

Is it true that $R=k[x,y]$ is a free $R$-module ? I think that it isn't true. Natural candidate for the base is $\{x^{\alpha}y^{\beta}\}_{\alpha,\beta}$, but : $x\cdot (xy) +(-y)\cdot x^2 =0$ and ...
1
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1answer
20 views

Find the integral closure of an integral domain in its field of fractions [duplicate]

Let $k$ be a field and let $R = k[x,y]/(x^2-y^2+y^3)$. Note that $R$ is an integral domain. Let $F$ be the field of fractions of $R$. How to determine the integral closure of $R$ in $F$? I have ...
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10 views

Find how many elements [on hold]

Find how many elements in a group of order 30 has the order 5,and explain the reasons. Cant do it. Any ideas?
2
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1answer
63 views

Finding a binary operation on $\{1, \dots, n\}$ so that each $k$ has exactly $k - 1$ left inverses

What is an example of a binary operation on the set $\{1, \dots, n\}$ so that each element $k \in \{1, \dots, n\}$ has respectively $k-1$ left inverses? I have been trying various combinations ...
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1answer
17 views

Question in proof from James Milne's Algebraic Number Theory

I'm having difficulty understanding a step in a proof from J.S. Milne's Algebraic Number Theory (link). Here $\zeta$ is a $p$th root of unity and $\mathfrak p = (1-\zeta^i)$ for any $1\leq i\leq p-1$ ...
6
votes
1answer
126 views

What exactly is Hensel doing for us in this result?

I'm reading a paper where the author appeals to Hensel's lemma, but it is not clear to me quite how it is meant to be applied (or, for that matter, which version!). My commutative algebra background ...
2
votes
1answer
54 views

Example of subgroup of $\mathbb Q$ which is not finitely generated

I was looking for a proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation. We know every finitely generated subgroup of $\mathbb Q$ is cyclic. For a proper ...
3
votes
2answers
71 views

Find all of the homomorphisms $ \varphi: \mathbb{Z}_{15} \to \mathbb{Z}_{6} $.

I'd like to find all of the homomorphisms $ \varphi: \mathbb{Z}_{15} \to \mathbb{Z}_{6} $. What I've tried so far: I know that $ |\text{Im} (\varphi)| $ divides $ ...
2
votes
1answer
46 views

Basic question on the free algebra

Let $k$ be a field and consider the (unital and associative) free algebra on $k$ with two generators ($x$ and $y$), $A= k < x,y >$. I have two basic questions concerning this algebra: 1) If ...