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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Is this theorem provable using relatively elementary number theory and abstract algebra?

$\textbf{Theorem}$: Let $p$ be a prime. Let $q$ be a prime that doesn't divide $p - 1$, so that $\mathbb{F}_p$ does have an element of order $q$. Let $\zeta$ be an imaginary number whose order is $q$. ...
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48 views

$[K(a):F(a)]=[K:F]$ if $a$ is transcendental over $K$.

Let $F$ and $K$ be subfields of the complex number $\mathbb{C}$ such that $K$ is a finite extension of $F$. Let $a\in \mathbb{C}$. If $a$ is not algebraic over $K$, prove that $[K(a):F(a)]=[K:F]$. I ...
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Can these two quotient groups be isomorphic?

Let $N$ and $M$ be two normal subgroups of a group $G$. Then we can show that the set $NM \colon= \{\ nm \ | \ n \in N, \ m \in M \ \}$ is a subgroup of $G$, that $M$ is normal in $NM$, and that $N ...
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35 views

Is every normal subgroup the kernel of some endomorphism?

Let $G$ be a group, and let $N$ be a normal subgroup of $G$. Then there is the canonical homomorphism $\phi$ of $G$ onto $G/N$ with kernel $N$. This homomorphism is defined as follows: $\phi(g) ...
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20 views

Questions on proof that transvections are conjugate in $GL(V)$.

I have difficulty following the proof that transvections are conjugates in $GL(V)$, and for $n \ge 3$ even in $SL(V)$. I give the necessary definitions and the proof, with the problematic parts ...
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1answer
63 views

If $xy=x^{-1}y^{-1}$, does this imply $x=x^{-1}$

This seems like a simple enough question, trying to show that if the title condition holds, that a group $G$ of which $x,y$ are elements, then $G$ is Abelian. $$xy=x^{-1}y^{-1}=(yx)^{-1}$$ From here I ...
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The definition “module of finite type”.

I know the definition of "module of finite type" Is that different from finitely generated module ? Thanks.
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34 views

Groups with single conjugacy class of subgroups. [duplicate]

I wish to know all those groups in which there is single conjugacy class of subgroups of fixed order.For example, In finite cyclic groups and in Alternating group of degree 4, number of conjugacy ...
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36 views

Find all sub groups of order $4$ in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$ . Are they all cyclic?

Find all sub groups of order 4 in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$. Solution : $\mathbf{Z}_4 =\{0,1,2,3\}$ $O(1) = O(3) = 4$, $O(0) = 1$, $O(2) = 2$ Hence, I found the subgroups of order 4 as ...
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action of symmetry group of cube on pairs of opposite faces.

I want to solve the following problem from Dummit & Foote's Abstract Algebra: Explain why the action of the group of rigid motions of a cube on the set of three pairs of opposite faces is not ...
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104 views

A group with finitely many subgroups must be a finite group

Show that a group that has only a finite number of subgroups must be a finite group.(Fraleigh, A First Course in abstract Algebra-7th Edition,pg.67) I could not show properly so I need help. Thank ...
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65 views
+50

Equivalence of definitions for “normal extension” and how to lift isomorphisms to them

Briefly: I want to prove that these two definitions for "normal extension" are equivalent: "$K$ is a splitting field for a collection of polynomials in $F[x]$" vs. "Every irreducible polynomial in ...
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21 views

as $n$ will be odd or even , $A_n $ generate with $(123)$ and $ (12…n)$ or with $(123)$ or $(23..n)$ .

as $n$ will be odd or even , $A_n $ generate with $(123)$ and $ (12...n)$ or with $(123)$ or $(23..n)$ . can you please help me to solve this problem.any hint or guide or reference will be ...
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1answer
62 views

The splitting fields of two irreducible polynomials over $Z / p Z$ both of degree 2 are isomorphic

$p$ is a prime. Let $ f_1, f_2 \in Z / p Z [t]$ both of degree 2 and irreducible. Show that they have isomorphic splitting fields. My approach was let $ K_1 = F(\alpha_1, \beta_1) / F$ be the ...
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1answer
43 views

Automorphism that maps primitive roots of unity.

Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ ...
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1answer
61 views

Failure of Luroth's theorem for transcendence degree 3

Can somebody give an example which shows the failure of Luroth's theorem for transcendence degree 3 over $\mathbb{C}$
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41 views

Irreducibility of a Polynomial over Q

How do I show that for any odd prime $p$ the polynomial $f(x)=x^p-9$ is irreducible over $\mathbb Q$?
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24 views

Cartesian Product of Sets and the Direct Product of Groups

I'm having a bit of confusion. I've tried to search youtube and whatnot but I could not find any explanations. My book says the following: The Cartesian Product is denoted by: $$S_1\times ...
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27 views

some basic concepts on algebra / measure theory

I'm reading a book in Chinese on measure theory (Introduction on Measure Theory, by Yan Jia-an). In the beginning there are some algebra concepts defined that I'd like to confirm the exact meaning and ...
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1answer
18 views

If $l(a, b, c) = l(a', b', c')$, then $(a, b, c) = (a ', b', c')k$ for some $k \in F$?

Let $F$ be a division ring. Define $l(a, b, c) = \{(x, y, z) \in F^3 : xa + yb + cz = 0\}$. Question: If $l(a, b, c) = l(a', b', c')$ is it true that $(a, b, c) = (a', b', c')k$ for some $ k \in F$? ...
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1answer
43 views

Product of Differences of nth Roots of Unity

I'm trying to show that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ij/n}\right)=n$$ but am finding it surprisingly difficult. I know by symmetry that $$\prod_{j=1}^{n-1}\left(1-e^{2\pi ...
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32 views

How do i prove this? (Cauchy's theorem)

Dummit-Foote p.96 Exercise 9 Let $G$ be a finite group divisible by a prime $p$. Define $S=\{(x_1,\cdots,x_p)\in G^n: x_1\cdots x_n = 1\}$ THen, $S$ has $|G|^{p-1}$ elements. HOw do i ...
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$|G| \geq 3 $ is finite and has subgroup with index more than 4 then $G$ is not simple.

suppose $G$ is a finite group with more than three elements,prove that if $H$ will be a proper subgroup of $G$ with this property that $[G:H] \geq 4$ then $G$ is not simple. I found this question a ...
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1answer
38 views

How do you find the automorphism?

How exactly would you find all the automorphism of something like $Z_8$ or $U(8)$? I read that there are $4$ automorphisms of $Z_8$, but how did they come about it? Please explain this as if ...
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1answer
57 views

Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
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42 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
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1answer
36 views

Show that the image or the kernel are submodule of R-module.

Let $R$ be Commutative ring and $M$ be an $R$-module. Show that $im(H)$ or $ker(H)$ are submodule of $R$-module $M$, where $$H\in Hom(M,-)$$ First, I think by lemma: If $M$ is an $R$-module and $N$ is ...
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33 views

Comparing coefficients in finite field

We start with the wrong proof of the following theorem: $p| \binom{p}{k}$ for a prime $p$ and $0<k<p.$ Proof: $(1+x)^p \equiv 1+x \equiv 1+x^p \pmod{p}$ by Fermat's little theorem. Comparing ...
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47 views

ring and module problem

Let $$F=\mathbb{R}$$ $$V=\mathbb{R}^{4}$$ consider two matrices $$S1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 ...
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1answer
16 views

Showing that a subrepresentation generated by an element is actually a subrepresentation.

Let $G$ be a group and $V$ be a representation of $G$. For $v_0 \in V$, the subrepresentation of $V$ generated by $v_0$ is constructed as $\{g \cdot v_0 | g \in G\}$. However, I don't immediately ...
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15 views

Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
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1answer
20 views

What do $a,b$ designate here?

Dummit-Foote p.76 Let $\phi:G\rightarrow K$ be a homomorphism. Let $G/\ker(\phi)$ be the set of fibers of $\phi$. Define $\phi^{-1}(a)+\phi^{-1}(b)\triangleq \phi^{-1}(ab)$ Then, ...
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1answer
30 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
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25 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
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27 views

Solving a system of modular equatios

Edit: I can't actually see how Chinese remainder theorem works here, if we had only $x$ on the left of each equation I can see how I could work it, but we don't. I can't seem to reduce it down to just ...
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Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
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Pro-p-groups and free pro-$p$-groups, an exact sequence (Generators and Relations)

I currently have the following issue (generators and relations for profinite groups): Let $\mathcal{S}={(g_i)}_{i\in I}$ be a system of generators for a pro-$p$-group $G$, then we can obtain an exact ...
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1answer
39 views

How to prove that it is a group?

Let $G$ be a set with a binary operation *, associating to each pair of elements $x$ and $y$ of $G$ a third element $x*y$ of $G$. Suppose that the following properties are satisfied: $(x*y)*z = ...
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number of distinct element

I have this assignment question that was given regarding: A group of permutation on a (let say) X number of letters having (let us also say) Y distinct element.... my problem is, how do I find the ...
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If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
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Finding Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$. The mistake in this method?

We need to find the Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$ . This method does not give me the right answer (i.e $6$ ) . Attempt: We need to find the number of Cyclic ...
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1answer
34 views

Maps from $SO(3)$ to $S_1, S_2$, and $S_1 \times S_2$

I am looking for continuous maps between the special orthogonal group of 3x3 matrices and the unit circle, unit sphere, and their product (S1, S2, S2 x S3, respectively). Any hints as to what I should ...
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1answer
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How to find the inverse of a polynomial modulo another polynomial?

how would you find the inverse of 2x+1 for modulus x^2+x+2? I did euclidean division, but I'm still rather confused x^2+x+2=(2x+1(2x)+(2x+2), 2x+2=4(5x)+2 5x=2(4x) all in F3 I know I'm wrong but ...
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1answer
24 views

Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
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32 views

What would the units and zero divisors be?

in F3[x] for x^2-1 is x a unit or zero divisor? I was wondering what the units and zero divisors would be would they be 0,1,2,x+1,x-1,2x+2,2x-2 and then units x-2,x+2 HELP
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Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
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59 views

A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$ where $|a|=m_0$ and $|b| = m$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some ...
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1answer
25 views

Any transformation in $A(V)$ satisfies a polynomial of degree $2$ over $\mathbb F$. [closed]

$V$ is a two-dimensional vector space over a field $\mathbb F$; prove that every element in $A(V)$, the ring of linear transformations on $V$, satisfies a polynomial of degree $2$ over $\mathbb F$.
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Elementary Proof is isomorphism between quotient groups of gaussian integers

In order to show that, for example $Z[i]/(2-i)\cong Z/5Z$ or $Z[i]/(4-i)\cong Z/17Z$, is there any solution that explicitly constructs a homomorphism between the two sets, establishes that it is a ...
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2answers
34 views

Methods for finding Number of roots in $\mathbb{Q}_{p}$ for a polynomial

The problem is: find how many roots in $Q_{p}$ does $x^{3}+25x^{2}+x-9$ have for p=2,3,5,7. I found for $\mathbb{Z}_{p}$. Is there a way to extend from here? Also, can you suggest me a book or link ...