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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Small categories

Why $R$-Mod is a small category? There is a way to recognize small categories? For example Grp (i.e. category of all groups) is large because every set can be equiped with a group structure.
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Following groups are not pairwise isomorphic in spite of having the same order? [on hold]

Can someone help me with the following question? "Prove (Z8, +), (D4, ◦) (the group of symmetries of the square) and the quaternion group (Q, ·): Q = {1, −1, i, −i, j, −j, k, −k} are not pairwise ...
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3answers
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How can I show that $G$ is non abelian of order 20?

Problem says: Let $G=\langle x,y|x^4=y^5=1,x^{-1}yx=y^2\rangle$. Show that $G$ is nonabelian group of order 20. To show it, I tried to turn $x^ny^m$ into $y^kx^l$ for some $k,l$. Since I have ...
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0answers
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Tensor product of complexes

Let $A$ be a ring and let the modules that are involved be left and right $A$-modules (not necessarily bimodules over $A$). I'll denote as $\mathcal{E}^n_R(M, N)$ the category of $n$-fold extensions ...
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1answer
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For $H \leq G$, showing that $N_G(H)/C_G(H) \leq \text{Aut}(H)$

This question probably has a very simple answer! I'm trying to understand the proof of the following result from Dummit and Foote, 3ed: Here is the proposition referenced: I don't understand ...
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1answer
70 views

Is this true: $A \le N_G(B) \not \Rightarrow B \trianglelefteq A$?

Let $G$ be a group, let $A, B$ be subgroups of $G$, and assume $A \le N_G(B)$. My question comes from reading Dummit and Foote, $\S 3.3$: The Isomorphism Theorems. We are proving the Second/Diamond ...
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1answer
44 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
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0answers
26 views

Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely: Given any ...
70
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2answers
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The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
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2answers
37 views

Artinian - Noetherian rings and modules suggest study guide

What text or any document that has gathered this part of Algebra theory. Thanks. Pd: I seek on variety's book of commutative algebra but the subject is partially dealt
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1answer
16 views

Is a ring R, modulo an ideal I (generated by x), then modulo an ideal J (generated by n) the same as R modulo the ideal generated by (n,x)?

Is the following statement true? $$ R/(x,n) = \left[ R/(x) \right] / (n) $$ My thinking behind it was as follows: \begin{array}{ccc} \left[ R/(x) \right] / (n) & = & \{ r+(n) : r \in R/(x) ...
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1answer
28 views

Onto Group Homomorphism S3 to K4

Problem: Determine if there exists an onto group homomorphism $\alpha: S_3 \rightarrow K_4$ (the Klein Group) Let K be the kernel of alpha. Here is what I have so far: Since any onto homomorphism ...
3
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2answers
551 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
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2answers
19 views

First isomorphism Theorem and Cosets of K

let $\alpha:G \rightarrow G_1 $ be a group homomorphism with ker $\alpha$ = K. For $a \in G$ show that Ka = {$g \in G$|$\alpha(g) = \alpha(a)$}. I am studying the first isomorphism chapter of my book ...
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0answers
30 views

Fundamental theorem of finitely generated abelian groups query

Just had a question about how to apply this theorem. If I am only told that a group is of finite order and is Abelian can we use this theorem? Is there a way to ensure it is finitely generated, or do ...
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1answer
33 views

$G = \mathbb{Q} / \mathbb{Z}$ surjective map and kernel isomorphism

Let $G = \mathbb{Q} / \mathbb{Z}$, written additively. For all $n > 0$ how come $p_n(x) = nx$ is a surjective homomorphism from $G \rightarrow G$ and how come the kernel of $p_n(x)$ is isomorphic ...
11
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1answer
219 views

Can we always find a primitive element that is a square?

Let $L/\mathbb Q$ be a finite field extension. The Primitive Element Theorem says that there is an element $\alpha \in L$ so that $L=\mathbb Q(\alpha)$. Can I always find an element $\beta \in L$ so ...
2
votes
1answer
22 views

Isomorphism of Subgroup of $D_n$

If $k|n$ , $k \ge 2$ I am trying to show that $D_n$ has a subgroup isomorphic to $D_k$. I know that by Lagrange's theorem If a subgroup of order $k$ exists in $D_n$ it will divide $2n$ then $k$ ...
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1answer
24 views

Quick way to show that inclusion is a local property? [duplicate]

I have encountered a problem which requires me to prove that ideal inclusion is a local property. That is to say, suppose $S,T \subset R$. Show that $S \subset T $ if and only if $SR_P \subset SR_P$ ...
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2answers
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For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Let us define numbers $a_i\in\mathbb{R}_{\geq0}$ by $a_0=0$ and $a_{i+1}=\sqrt{2+a_i}$. How do I prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$? How do I prove that ...
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Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
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Galois theory for beginners, mistake?

I read this short article http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf. The goal of this article is to show the unsolvability of certain polynomials of degree $\geq 5$. But at the ...
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3answers
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If M+N and M$\cap$N are finitely generated modules, so are M and N.

The question asks to prove that if $M+N$ and $M \cap N$ are finitely generated modules, then M and N are also finitely generated. I've tried to use basic definitions, but all failed. I set some ...
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2answers
37 views

$M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences [duplicate]

Let $0 \to M_1 \to M \to M_2 \to 0$ be an exact sequence of $A$-modules. i) Prove: If $M_1$ and $M_2$ are finitely generated, then $M$ is too. ii) Let $M$ and $N$ be sub-modules of an $A$-module ...
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1answer
26 views

Show that $|\Bbb Z_n^*|$ is even if $n\ge 3$.

I am trying to show that $|\Bbb Z_n^*|$ (i.e the elements of $\Bbb Z_n$ relatively prime to n forming a group under multiplication) is even if $n \ge 3$ using a corollary in my textbook to Lagrange's ...
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1answer
25 views

Submodules finitely generated if sum and intersection are finitely generated? [duplicate]

Let $M,N$ be submodules of an R-module $L$, where $R$ is a commutative ring with unity. I would like to prove that if $M+N$ and $M \cap N$ are finitely generated so are $M$ and $N$. Trying to combine ...
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2answers
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Fibers of extension of scalars to algebraic closure of an affine variety is of dimension 0.

Let $X$ be an affine variety over $k$ and $\overline{k}$ be the algebraic closure of $k$. Then the projection morphism $X_\overline{k} = X \times _k \overline{k} \to X$ has dimension 0 fibers. ...
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4answers
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Proving that $G/N$ is an abelian group

Let $G$ be the group of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}$ where $ad \neq 0$ under matrix multiplication. Let $N=\left\{A \in G \; \colon ...
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2answers
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$R/\Bbb Z$ isomorphic to $R/(2\pi \Bbb Z)$

I was told that $\mathbb{R}$$/$$\mathbb{Z}$ is isomorphic to $\mathbb{R}/2\pi \mathbb{Z}$ when these groups are taken under addition. Is this always true? I do not specifically see why this has to be ...
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0answers
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Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
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2answers
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$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing I know which connects a group ...
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1answer
22 views

Every subgroup of finite index contained in an infinite group $G$ contains a normal subgroup of $G$. [duplicate]

Let $H<G$ such that $H$ has finite index in the infinite group $G$. Then $G$ contains a normal subgroup of finite index contained in $H$. Can I create a subgroup of index $2$ in $G$ using elements ...
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2answers
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Finding the orbits of the orthogonal group $O(n)$ on $\Bbb R^n$

Let $O(n)=\{M\in GL_n(\mathbb{R}):MM^t=M^tM=I\}$ an orthogonal group. I need please an explain why each orbits consists of all vectors with the same length. I know that an orbit is defined by ...
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2answers
39 views

Principal Ideal using coordinates?

I thought I understood principal ideals but now im stuck... I want to find the elements of the principal ideal $\langle(1,0)\rangle$ in the ring $\mathbb Z_3\times \mathbb Z_3$ with $+_3$ and $*_3$ in ...
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1answer
30 views

Finding exact isomorphism between finite fields given as quotient rings [duplicate]

I have two quotient rings over $\Bbb F := GF(3)$: $$\Bbb F[x] / (x^3 -x - 1) \qquad \text{and} \qquad \Bbb F[x] / (x^3 -x + 1) .$$ These things I know: Both quotient rings are irreducible, that means ...
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1answer
31 views

A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
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3answers
533 views

$A_4 \oplus Z_3$ has no subgroup of order 18

Here my solution: Suppose there exists and $H \leq A_4 \oplus Z_3$ such that order of H is 18. Now, notice index of H in $A_4 \oplus Z_3$ is 2. therefore, H is normal, and therefore, the $A_4 \oplus ...
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1answer
20 views

Image and Kernel of composition of two homomorphisms

I have just showed that the composition of $a * b$ of two homomorphisms $a,b$ is a homomorphism. However, what can I say about the image and kernel of $a*b$, in terms of $a$ and $b$? Is there ...
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3answers
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Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.

Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements. We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$. If $G$ is ...
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1answer
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Dimension about space of matrices of order 3 over the field of the real.

Consider the vector space of the matrices of order 3 over the field of the real $M_{3}\left(\Re\right)$ numbers. and let S be the subspace such that is spanned by the matrices of the form $AB-BA$. ...
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1answer
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How many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads?

I am trying to figure out how many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads. I understand how to do the problem with two colors, but I am struggling to ...
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1answer
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Is it possible to say that if a A is a set of elements of a semigroup S, there exists some other semigroups on the alphabet A?

Suppose we have a finite semigroup, its element's set is A = {1,2,...n}, can we get some other semigroups whose element's set is A too ?
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proof that $\frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1}$ is irreducible

I am reading the group theory text of Eugene Dickson. Theorem 33 shows this polynomial is irreducible $$ \frac{x^p - 1}{x-1} = 1 + x + \dots + x^{p-1} \in \mathbb{Z}[x]$$ He shows this polynomial ...
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Showing $\mathbb{Q} \times \mathbb{Q}$ is not a field

I am revising and have come across the question Show that $\mathbb{Q} \times \mathbb{Q}$ with element-wise addition and multiplication is not a field I don't understand how to go about this, do i ...
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1answer
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Extending a given element of a free abelian group to a basis

Let $V$ be a free $\mathbb{Z}$-module (or a free abelian group) with basis $(v_1$, $v_2$, $v_3)$. Denote by $x$ an arbitrary nonzero element of $V$, say $x=mv_1+nv_2+kv_3 \in V \setminus \{0\}$; here ...
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2answers
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How to prove that two principal ideals are equal [on hold]

For $F$ a field, and $q(x)$ a polynomial in the polynomial ring $F[X]$, with $a\in F$ where $a \neq 0$, show that $\langle q(x) \rangle= \langle aq(x) \rangle$, where $\langle \cdot \rangle$ denotes ...
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discrete valuation ring for formal series

as I can define a discrete valuation ring for formal series. I can define it as: Formal power series ring, norm.
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Primitive element in multivariate Galois field [on hold]

On Singular CAS I can define a Galois field $(2^3)$ with $(x,y,z)$ variables. But I am not able to understand how $a^3+a+1$ is still its primitive element. General example taken in books is always ...
3
votes
2answers
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Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
4
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1answer
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What are the natural surjections in the proof of Hopf's classification theorem?

I am currently reading Hatcher's book, trying to understand the proof of Hopf's classification Theorem on Hopf algebras that says the following: Every Hopf algebra $A$ that is commutative and ...