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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Explicit formula for a multiplication

Do you know any explicit formula for the multiplication: $$(x-x_1)(x-x_2) ...(x-x_n)$$ It should be an explicit easy formula, but I cannot find it ...
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0answers
33 views

Basic proof in algebra linear [closed]

I have a problem with: We have a in F -field and a is any element of F. I need proof that ...
3
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3answers
43 views

Finding $\text{Aut}(D_3)$

The following is a problem from my algebra homework: Find all automorphisms of $D_3$ and determine which group $\text{Aut}(D_3)$ is isomorphic to. I am fairly new at abstract algebra so this problem ...
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2answers
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Prove or disprove that $G_1/H_1 \cong G_2/H_2$

Let $\phi : G_1 \rightarrow G_2$ be a surjective group homomorphism. Let $H_1$ be a normal subgroup of $G_1$ and suppose that $\phi (H_1) = H_2$. Prove or disprove that $G_1/H_1 \cong G_2/H_2$. I say ...
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Alternate Form of Correspondence Principle

Let G be a group and K a normal subgroup in G. Let f : G → G/K be the canonical epimorphism given by x 􏰁→ xK and L a subgroup of G/K. Then (1) There exists a subgroup H of G. H contains K, and ...
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Exhibiting triviality in the Brauer group

The starting place for this question lies in the following construction: given a field $F$ containing a primitive $n$th root of unity $\omega$ and two elements $\alpha, \beta \in F^{\times}$, we ...
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1answer
39 views

Writing a particular polynomial as product of irreducibles in various rings.

I want to factor the polynomial $x^3-10x+4$ into a product of irreducibles over each of the fields $\mathbb{Z}[i]$,$\mathbb{Q}[\sqrt{2}]$, $\mathbb{Q}[\sqrt{2},\sqrt[3]{2}]$, $\mathbb{Z}/ 11 ...
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1answer
30 views

Finding all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$

I want to find all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$. I know that $$\mathbb{Q}[x]/I \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x^2+x+1)$$ I ...
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1answer
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Irreducibles but not primes in $K[X,Y]/(X^2-Y^3)$

I'd like to know why $X+(X^2-Y^3)$ and $Y+(X^2-Y^3)$ are irreducible, but not prime in $K[X,Y]/(X^2-Y^3)$. I failed in both. For the first, I tried to use an isomorphism ...
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1answer
24 views

Orbits of General Linear Group

I am working on a problem to find the orbits of the general linear group ${GL_n(R)}$, acting on ${R^n}$, with the invertible matrix A acting on a column vector x in ${R^n}$ by taking it to the vector ...
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2answers
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Groups of order 24.

I supposed $n_3=4$ and $n_2=3$, and then I made $G$ act by conjugation on $Syl_3 (G)$. I want to show that $G\cong S_4$ (looking at all order 24 groups here ...
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29 views

Orbits of ${S_n}$ in ${J \times J}$

I am trying to show that for ${S_n}$, there are exactly two orbits in ${J \times J}$, where $J = {1,2,....n}$. My proof so far proceeds as follows: Take ${(x,y) , (a,b)}$ in $J \times J$. Take s in ...
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2answers
51 views

Identities about ideals

If $A,B \subseteq K^n$ show the following: If $A \subseteq B$, then $I(B) \subseteq I(A)$. $I(A \cup B)=I(A) \cap I(B)$ Could you give me a hint, how the above identities could be proven? EDIT: ...
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1answer
37 views

Showing that $\mathbb{R}^2 /N \overset{\simeq}\to g_0$ where $g_0$ denotes a line

Consider in the $\mathbb{R}$-Vectorspace $\mathbb{R}^2$ the sub vectorspace $N:= \mathbb{R}w$ where $w\neq 0$. For $w_0 \neq 0$ let $g_0:= \mathbb{R}w_0 \subset \mathbb{R}^2$ be the sub vectorspace ...
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Tools for understanding Abstract Mathematics

I want to explore the concepts of Groups, Rings and Fields for practical application. Are there beginner-friendly tools available to simplify, accelerate, verify the understanding of concepts in ...
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28 views

Partial generalisation to Whitehead's second Lemma

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional semisimple $k$-Lie algebra. By Whitehead's second Lemma, we know that $H^{2}(\mathfrak{g}, ...
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1answer
36 views

A commutator relation

I hope the following is not trivial, Let $H$ be a subgroup of $G$ s.t. $[H,G]\leq Z(G)$ then can we say that $H$ is normal ? I think we can not but I could not find counter example. Any counter ...
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1answer
57 views

Is $R$ PID if every submodule of a free $R$-module is free?

Let $R$ be a commutative ring. Before I proved that every submodule of a free $R$-module is free over a P.I.D. Now I'm trying to prove the reciprocal, if every submodule of a free $R$-module is ...
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0answers
34 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
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questions about proof of existence of clifford-algebra

Let $C$ be a Clifford-algebra, e.g. $(V,q)$ is a quadratic space, $C$ an associative algebra with $1$, $j:V\to C$ linear and $j^2(v)=q(v)\cdot 1$ for all $v\in V$, and if we have another associative ...
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1answer
44 views

If both N and G/N are cyclic, then G is cyclic. [duplicate]

Prove or disprove that if both N and G/N are cyclic, then G is cyclic. N is a normal subgroup of a group G. I think it is false but I can't find a counter example.
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The sum of the elements in a field of at least three elements is 0

This statement seems so simple yet I don't quite know how to start with this proof in a substantial way. Can anybody help me here?
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47 views

Nilpotent torsion-free Groups with a fixed Soluble length

Let $s$ be a natural number. Is it possible to find, for each $n$ natural number greater than some arbitrary constant, a torsion-free group whose nilpotency class is $n$ and soluble length is $s$?
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Ideal in the ring of quotients

I got some problems with the following: Let $R$ be a ring with $1\in R$. Let $S\subset R$ a subset which is closed under multiplication and contains $1$. On the set $R \times S$ we define an ...
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1answer
32 views

Intuitive meaning of left and right cosets?

What is the difference between left and right cosets? I know their definition, but what I am seeking is the intuition behind left and right coset. I used to think of cosets as slicing a group (since ...
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Permutations in products of disjointed cycles

How do I calculate the following permutation in the symmetric group $S_6$ giving the answers as products of disjoint cycles: $$(2,3,5,6)(1,6,2,4)$$ I have tried following this question but I don't ...
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1answer
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Let H be a subgroup of a group G. Prove that the following statements are equivalent.

Let H be a subgroup of a group G. Prove that the following statements are equivalent. (a) For all $a,b \in G, (aH)(bH)$ is a left coset of $H$ in $G$. (b) For all $a,b \in G, (aH)(bH) = (ab)H$. (c) ...
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Existence of a Polynomial

Does their exist a non-linear polynomial $P(x)$ such that for every rational number $y$ there exists a rational number $x$ such that $y=P(x)$?
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In a group of symmetries, find elements such that $\sigma \circ\beta = \beta \circ\sigma$ and $\sigma \circ\beta = \beta^{-1} \circ\sigma$

(i) Let $\sigma = (12)(345) \in S_5$. Find all $\beta \in S_5$ such that $\sigma \circ\beta = \beta \circ\sigma$. $S_5$ is the symmetric group of 5 elements. (ii) Let $\sigma \in S_n$. Show that ...
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If n $\geq2$, does G necessarily have an element of order $p^2$? Justify your answer. [closed]

If n $\geq2$, does a group G of order $p^n$ necessarily have an element of order $p^2$? Justify your answer. p is a prime number. n is an integer. I can't seem to find a counterexample, but I think ...
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1answer
51 views

Prime avoidance lemma [closed]

Let $R$ be a commutative ring. $A$ is an ideal of $R$, and $P_1,P_2,...,P_s$ are prime ideals of $R$, $$A \subset \bigcup_{j=1}^{s} P_j$$ Prove that there exists some $P_j$ such that A is contained ...
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1answer
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Question on the proof of Cauchy Theorem

This is the proof for the general case given in wikipedia. In the general case, let Z be the center of G, which is an abelian subgroup. If p divides |Z|, then Z contains an element of order p by the ...
2
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1answer
58 views

Proof of the First Isomorphism Theorem for Rings [duplicate]

The statement is the First Isomorphism Theorem for Rings from Abstract Algebra by Dummit and Foote. I'd like to check if all is ok. In particular I'm a bit worried about the (*) line. It looks a bit ...
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1answer
63 views

Proof of the First Isomorphism Theorem for Groups

The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit ...
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Showing well-definedness of a group operation

Let p be an odd prime integer, and let $r \in \mathbb{Z}$ with $1 \leq r < p$. Let G be a cyclic group of order p generated by g, and let K be a cyclic group of order p-1 generated by k. For ...
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1answer
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Examples of modules (basis and generators)

I have to give examples for modules such that $1.$ There's a linearly independent set with more elements than a generator set. $2.$ There's a linearly independent set with the same cardinal than a ...
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1answer
22 views

Show that a certain normal subgroup or a product is abelian

Let $A$ be a normal subgroup of $G\times H$. If the identity of $G$ is $1_G$ and the identity of $H$ is $1_H$, $(x,y)\in A$ has the property that $x\ne1_G$ and $y\ne1_H$ unless $(x,y)=(1_G,1_H)=1_A$. ...
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1answer
33 views

Proving Product rule with Abstract Algebra Methods

Theorem: Let $F$ be a field and $p(x) = a_nx^n+a_{n-1}x^{n-1} + \ldots + a_0$ be a polynomial in $F[x]$. The derivative of $p(x)$ is the polynomial $ D(p(x)) = na_nx^{n-1}+(n-1)a_{n-1}x^{n-2} + ...
3
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1answer
39 views

Maximal ideal in $K[x_1,…,x_n]$ [duplicate]

I'm having some difficulty with this homework problem: If $A=K[x_1,...,x_n]$, $K$ a field and $a_1,a_2,...,a_n $ $\in K$. The ideal $m=<x_1-a_1,...,x_n-a_n>$ is maximal.
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Irreducible representations of $S_3$

I am trying to do a problem from artin's algebra 2nd ed (Chapter 10, Exercise 2.3) but having trouble: Let $(\rho , V)$ be a representation of the symmetric group $S_3$. Let $x=(123), y=(12)$ be the ...
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Cardinal of a linearly independent subset of $R$-module

Let $R$ be a commutative ring, and consider $R$ as an $R$-module with the action given by the product of $R$. Prove that if $B\subset R$ is linearly independent, then $\operatorname{card}(B)=1.$ ...
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Is it true that any quotient group G/N is abelian if N contains the commutator subgroup? [duplicate]

Is it true that any quotient group G/N is abelian if N contains the commutator subgroup(including the case when N is the commutator subgroup)? I think this is true but I just want to confirm.
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Good resources to learn Hopf algebras

I am finding it hard to find resources that can educate me in this subject. I have looked up this and I can't find anywhere to start. Any advice?
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38 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
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2answers
85 views

Order of group from its presentation

Say we want to determine the order of a group generated by $x$ and $y$ who satisfy $x^2y = xy^3 = 1$. Ok so it would be nice to know the order of $x$ and $y$ respectively. We can readily conclude ...
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1answer
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Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
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13 views

Trace Map and Solution to Equation

Consider the equation $x^{31} - 1 =0$. Determine the number of solutions $\gamma \in \mathbb{F}_{1024}$ to the equation that satisfies $Tr(\gamma)=0$. I did some research on properties of field ...
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1answer
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Splitting field of $x^ {a^n}$ −1 in Z/aZ[x]

What is the splitting field of the polynomial $x^ {a^n}$ −1 in \ the ring Z/aZ[x] with n natural? I´m working in some kind of proof of the best known theorem that says it´s impossible there exist a ...
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K[x] module is k vector space

Could anyone tell me why the A-module is a k- vector space with a linear transformation? Here A is k[x] where k is a field. Thanks!
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Show this is an isomorphism.

I am trying to show that the mapping from $G \rightarrow G/H +G/K$, where + is the direct product is a homomorphism. The mapping is defined as $\phi(g)=(gH,gK)$. I have also showed that for the normal ...