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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Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? [on hold]

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? Does there exist a subgroup of $G$ of order $180$?
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6answers
1k views

If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic

Fraleigh(7th ed) Sec10, Ex47. Let $G$ be a finite group. Show that if for each positive integer $m$ the number of solutions $x$ of the equation $x^m=e$ in $G$ is at most $m$, then $G$ is cyclic. ...
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1answer
46 views

Asterisk Notation

I am trying to read through the Cech Cohomology section in these notes: http://pub.math.leidenuniv.nl/~edixhovensj/teaching/2011-2012/AAG/lecture_14.pdf I would be really grateful if someone could ...
5
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3answers
150 views

When is an element of a free module over a principal ideal domain contained in a basis?

I'm trying to show the following: Let $R$ be a principal ideal domain and let $M$ be free $R$-module of rank $n$. Let $Y=\{y_1,\ldots,y_n\}$ be a basis of $M$ and $x\in M$ with ...
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1answer
38 views

Irreducible action of a group on a set

Let $p$ be a prime. I am solving a problem and I am told that I should use that the action of $\text{SL}_2(p)$ on $\mathbb{F}_p^2$ is irreducible. But I don't know what this means?
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0answers
20 views

Lower central series and the center of $S/S^k$

Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. I don't know much about lower central ...
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1answer
42 views

Abelian group over a field underlying an abstract vector space [on hold]

Given that a set V is said to be a vector space over a field F if V is an Abelian group under addition and for each $a\in F$ and $\boldsymbol{v}$ in V there is an element $a\boldsymbol{v}$ in V, how ...
9
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5answers
298 views

Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the ...
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1answer
40 views

Number of monic irreducible polynomials over a finite field

Let $\mathbb{K}=\mathbb{F}_q$ and $\nu_n$ denote the number of monic irreducible polynomials over $\mathbb{K}$. It holds $$\nu_n=\frac{1}{n}\sum_{d\mid n}\mu\left(\frac{n}{d}\right)q^d$$ What I need ...
4
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1answer
55 views
+100

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
2
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2answers
67 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
1
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1answer
15 views

Simple $M_n(D)$-module with $D$ a division ring

Define $D$ to be a division algebra over a field $k$ and $R=M_n(D)$ the $n\times n$ matrix ring over $D$. A simple $R$-module $M$ is the quotient of $R$. I can write $R=\bigoplus_j I_j$ where $I_j$ is ...
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1answer
32 views

Resources for Polyadic and/or Cylindric Algebra

I'm looking to learn a little bit about polyadic and cylindric algebras, as part of an investigation into algebraic approaches to logic. The only "text" that I can find for polyadic algebra is ...
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0answers
28 views

Proving $R=\{a+b\sqrt3\mid a,b \in \Bbb Z\}$ is Euclidean. [on hold]

Let $R=\{a+b\sqrt3\mid a,b \in \Bbb Z\}$ A. Prove $R$ is a Euclidian domain with respct to the norm $N(a+b\sqrt3)= |a^2-3b^2|$. B. Divide $1+\sqrt3$ by $2+\sqrt3$ in $R$.
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0answers
29 views

simple exercise in Cylindric algebra

I am trying to gain a better understanding of cylindric algebra, so I made up this example. Given a general rule that someone's father's father is his/her grandfather: $\forall_X ~ \forall_Y ~ ...
0
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2answers
71 views

Is $2x^2+4$ reducible over $\mathbb C$?

I am not sure if I making some very fundamental mistake. But Gallian says that $2x^2+4$ is reducible over $\mathbb C$. If $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ is said to be ...
0
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2answers
51 views

May a monoid have two disjoint submonoids?

I'm asking this question inspired by the similar question about group and its subgroups. I tried to modify the proof presented there to work for monoids but I failed. I'm also not able to find any ...
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1answer
30 views

Findout if $\min(x,y)$ and $\max(x,y)$ are semigroup, monoid or group

I need to find out whether $A \oplus B := \max(A,B)$ or $A \ominus B := \min(A,B)$ form a semigroup, monoid or group for $\oplus : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ and $\ominus : ...
3
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2answers
102 views

Is an arbitrary group generated by a traversal of the conjugacy classes?

Let $G$ be a group, and let $\mathcal C$ be the collection of conjugacy classes of $G$. Let $S$ be a traversal of $\mathcal C$ (that is $S$ contains exactly one element from each set in $\cal C$). ...
2
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1answer
55 views

How many (unordered) bases does $\Bbb F_q^n$ have as a vector space over $\Bbb F_q$?

Following the recommendation here to get this question out of the unanswered queue, I've changed this from a proof-verification question into an answer-your-own. Here's the question again in case ...
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1answer
78 views

Linear algebraic group

Let A be a finite dimensional algebra over C . This means that there is a multiplication map $f : A \times A \to A$ that is bilinear ( it is not assumed to be associative). Define the automorphism ...
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1answer
72 views

The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the ...
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1answer
30 views

Quotient Gaussian Integers

Following Quotient ring of Gaussian integers, their extended conclusion is $\mathbb{Z}[i]/(a-ib) \cong \mathbb{Z}/(a^{2}+b^{2})\mathbb{Z}$. However it does not convince me, at least, one example ...
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2answers
30 views

R is a PID, and a is a nonzero nonunit in R. How can we show R/Ra is an injective module over R?

If we use Baer's criterion then it suffices to show that if there exist a map from an ideal $I$ to $R/Ra$ we must find a map $g$ such that $g\circ i=f$.
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6answers
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Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong ...
0
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1answer
18 views

Restricting Binary Operator $*$ To A Subset

I am currently reading Dummit and Foote's Abstract Algebra, and am having a little confusion over the following excerpt: Suppose that $*$ is a binary operation on a set $G$ and $H$ is a subset of ...
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1answer
44 views

Confused about proof in Basic abstract algebra by Bhattacharya

On page 264 , 2nd edition. Theorem 5.1 It says Let M be a free $R$-module with "a basis" $\{e_1,\dots,e_n\}$ Then $M$ is $R$-isomorphic to $R^n$. Above he is defining the standard basis as the ...
2
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45 views

Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
2
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1answer
41 views

Why are these dimensions equal?

For a finite $K$-algebra $A$ and $L\supset K$ fields, why do we have$$\dim_K A=\dim_L(A\otimes_KL)?$$ I ran across this a couple of times and it's always assumed to be quite obvious, which it isn't to ...
3
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1answer
48 views

Tensor product of duals

Can we show $V^* \otimes W^* \simeq (V \otimes W)^*$, when $V$ or $W$ is finite-dimensional without referring to the basis? I can inject $V^* \otimes W^*$ into $(V \otimes W)^*$ using the obvious ...
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0answers
26 views

Text on Witt vectors that are accessible to undergraduate students

I am looking for a thorough text on Witt vectors that is accessible to an undergraduate student that have completed the following courses: Calc 1, 2, Linear Algebra and Abstract Algebra. (In Norway, ...
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0answers
21 views

AI / Machine Learning related to high/modern/front mathematics [on hold]

I major math and cs. and i'm interested in ai/machine learning/data mining. so i want to know what math subjects are used in frontier of these technology. especially, high mathematical tool, like ...
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1answer
15 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
2
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1answer
41 views

Proving result on algebraically closed fields

I have been told that: Let $f_1,\dots,f_n,g\in F[x_1,\dots,x_m]$ be polynomials in $m$ variables with coefficients in the algebraically closed field $F$. Then if the system: ...
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1answer
15 views

Proving a certain set is inductive

Let's give some context. We have to prove «Krull's theorem», which states that: If $A$ is a commutative ring, $N$ is its (nil)radical, i.e. the set of its nilpotent elements $\{x\in A:\exists ...
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2answers
27 views

groups products vs vector space products

I started from wandering if the cross product (a product between two vectors that gives a vector) can be abstracted like dot product (a product between two vectors that gives a scalar) is abstracted ...
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0answers
31 views

Simple Cogenerator for the category of left $R$-modules

I want to prove that if the category of left $R$-modules has a simple cogenerator $M$, then $R$ is a simple ring. (Here, $R$ is an arbitrary ring with $1$.) My try is to take an ideal $I$ of $R$ ...
0
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1answer
18 views

proving closure of a subset [duplicate]

Let B be a set, and let * be a binary operation in B. Suppose * satisfies the associative law. Let $$P=\{b \in B : b * w = w * b \quad\forall\, w \in B\}$$ Prove that P is closed under *.
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1answer
26 views

Repeated Irreducible Representations in a representation

I'm reading through Serre's - Linear representations of finite groups. He has the following theorem (theorem 4 of chapter 2): Let $V$ be a linear representation of $G$, with character $\phi$ and ...
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1answer
34 views

Maps preserving roots of a polynomial function over finite fields

Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$. Let $S(P)=\{ ...
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1answer
43 views

Finitely generated, periodic group such that each conjugacy class is finite must be finite?

This is essentially a repeat of this question. However, the OP didn't seem to put in any work towards a solution and didn't provide context. Nevertheless, I'm still interested in the solution, and I ...
4
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1answer
140 views

Direct product of finitely many Noetherian non-unital rings is Noetherian

Let $A_1, A_2,...,A_n$ be Noetherian rings (not necessarily unital). Is the direct product $A:=A_1×A_2×⋯×A_n$ necessarily a Noetherian ring? If $A_1, A_2,...,A_n$ are unital, then one can prove ...
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0answers
30 views

Can the derived subgroup be realized as an intersection of stabilizers?

For any group $G$, we have that $Z(G)=\bigcap_{x\in G}C_G(x)$ and that each $C_G(x)$ is the stabilizer of $x$ when $G$ acts on itself by conjugation. Is there a similar representation for $G'$? That ...
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1answer
25 views

Finitely generated ring of polynomials

Can we say that, "by definition", a ring $R[x]$ is finitely generated as an $R$-module for some commutative ring $R$ iff $x^n=q(x)$ for some polynomial $q(x)$ of degree $n-1$ for some $n$?
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1answer
214 views

Finding the number of elements of order $2$ in the given group

How many elements of order $2$ are there in the group of order $16$ generated by $a$ and $b$ such that $o(a)=8$ and $o(b)=2$ and $bab^{-1}=a^{-1}$ I basic thing i do not understand is that order of ...
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1answer
66 views

Difference between algebraic and integral extension

I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below). Now I ...
2
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1answer
261 views

Homomorphism between symmetric group and general linear group of order n.

I am having trouble proving the following: Show that $f: S_n \to GL_n(\mathbb{R}),\;\: f(x)=A_x$ is a homomorphism where $A_x$ is the permutation matrix associated with $x$. $S_n$ is the ...
2
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1answer
57 views

The degree of the splitting field of $X^6+X^3+1$

Suppose $L \subset \mathbb{C}$ is a splitting field for the polynomial $X^6+X^3+1$ over $\mathbb{Q}$. Determine $[L:\mathbb{Q}]$ I have several solutions for the problem. However I'm having trouble ...
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0answers
17 views

number of group homomorphism from the symmetric group $S_3$ to the additive group $\Bbb Z/6\Bbb Z$ [duplicate]

The number of group homomorphism from the symmetric group $S_3$ to the additive group $\Bbb Z/6\Bbb Z$ ? a)1 b)2 c)3 d)0
-2
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0answers
26 views

Number of normal subgroups of a non-abelian group of order 21 [on hold]

How many normal subgroups can a non-abelian group G of order 21 have other than the identity subgroup {e} and G? a)0 b)1 c)3 d)7