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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Is the Kähler differential of a continuous function ring trivial?

Suppose $A=C^0(\mathbb R)$ is the ring of real-valued continuous functions on $\mathbb R$. Is it true that, the Kähler differential $\Omega_{A/\mathbb R}$ trivial? In other words, suppose that ...
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0answers
16 views

Solution space of semilinear equation

I found the following lemma and the corollary in a paper and I don't know how to prove them. Therefore I was wondering if one of you could help me. Let $E$ be a field, $ \sigma: E \rightarrow E$ ...
2
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1answer
32 views

Multiple correct question based on permutation group [duplicate]

I try to solve this the number of permutation in $S_{n}$ for $n\geq 4$ which are product of two disjoint $2$-cycles is $\frac{n(n-1)(n-2)(n-3)}{8}$. So after putting $n=5$ and $n=4$ I get different ...
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1answer
37 views

Inverses without identity possible?

Suppose $S$ is a nonempty set on which is defined a binary operation $*$ such that for all $x$, there exists a unique $y$, such that for all $z$, ...
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1answer
18 views

Visualising the absolute value homomorphism

In the Artin's "Algebra" book the fibres of the absolute value homomorphism $\phi: \mathbb{C}^\times \rightarrow \mathbb{R}^\times$ defined as $\phi(a) = \lvert a \rvert$ are visualised by concentric ...
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1answer
43 views

Product of sets as complexes

What does it mean to take the product of two sets of complex numbers as complexes? Reading this paper: "The Determinant of the Sum of Two Normal Matrices with Prescribed Eigenvalues" by N. Bebiano ...
7
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1answer
55 views

Counter-examples of Galois Correspondence

What are some examples of a separable field extension $L/K$ and a subgroup $H$ of $\text{Aut}(L/K)$ such that $\text{Aut}(L/L^H) \neq H$? Here $L^H$ means the fixed field of $H$.
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24 views

Show that $\varphi_{x,y}$ is a automorphism iff $\mathrm{ord}(x)=\mathrm{ord}(y)=n$ and $\langle x\rangle\cap\langle y\rangle=\{\hat{0},\hat{0}\}$.

Let $n\in\mathbb{N}$ and $$\varphi_{x,y}:\left(\mathbb{Z}/n\mathbb{Z},+\right)\to\left(\mathbb{Z}/n\mathbb{Z},+\right)$$ $$(\hat{m},\hat{l})\to m\cdot x+l\cdot y$$ a homomorphism. Show that ...
4
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1answer
44 views

Finite Almost Simple Groups

I want to study finite almost simple groups but I am not sure which would be the best texts to look at. Can someone please refer me to some books that teach the theory of finite almost simple groups?
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0answers
29 views

Is $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n\mathbb{Z}[G]$?

Let $G$ be a group, $\mathbb{Z}[G]$ be it's group ring and $G^n$ the direct product of $n$ copies of $G$. Is the group ring $\mathbb{Z}[G^n]$ isomorphic to $\bigoplus_n \mathbb{Z}[G]$? If not, is it a ...
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1answer
15 views

If $M$ is a simple $R$-module, and an $F$-space, why does $End_F(M)\cong M^{\oplus\dim_F(M)}$?

Suppose a ring $R$ is an $F$-algebra for $F$ a field, and $M$ is a simple $R$-module and a finite dimensional $F$-vector space. We can endow $\operatorname{End}_F(M)$ with an $R$-module structure by ...
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0answers
22 views

ADE classification of singular surfaces (catastrophe theory)

I have seen a lot the Arnold's classification of singular surfaces by the simple Lie groups. I have even asked the author of a book that used this classification about its origin and his answer was ...
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0answers
38 views

Algebra quotient space homomorphism

I have to prove the following; Let $A$ be an algebra over a field $K$. If $I \subset A$ is an ideal, then there exists a unique algebra structure on the quotient vector space $A/I$ such that the ...
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1answer
48 views

Ring theory. Find the isomorphism

With the help of the theorem of homomorphism for rings, find an isomorphism $\mathbb{Q} [x] / (x^2 - x) \simeq \mathbb{Q} \oplus \mathbb{Q}$, where $\mathbb{Q} \oplus \mathbb{Q} = \{ (q_1, q_2) \mid ...
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0answers
42 views

Find the dimension of the $\mathbb{R}$-algebra $\mathbb{R} [x] / (x^3 - x + 1)$ [closed]

Find the dimension of the $\mathbb{R}$-algebra $\mathbb{R} [x] / (x^3 - x + 1)$
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1answer
34 views

Find elements in the Ring [on hold]

Find all invertible elements, all divisors of zero and all nilpotent elements in the ring $R = \left\{ \begin{pmatrix} a & 0\\ b & c \end{pmatrix} \mid \, a, b, c \in \mathbb{R}\right\}$ with ...
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0answers
101 views

Number system where $e^x$ is identically zero [on hold]

In a previous post I tried to come up with an interesting example of a finite dimensional real algebra in which $e^x$ can be $0$. My attempt was not successful. I think part of the problem is that ...
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1answer
21 views

Normalizer of a quotient

Suppose that $G$ is a group and $A, B$ and $H$ are subgroups of $G$ with $B\unlhd A$. In a paper I have read, they say that the normalizer of $A/B$ in $H$ is $N_H(A/B) = N_H(A) \cap N_H(B)$. I am ...
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3answers
83 views

Number of Subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime

As the title says, I am interested in finding all subgroups of $C_p \times C_p$ and $C_{p^2}$ for $p$ prime. We did not cover the Sylow-theorem so far in the lecture. What I noticed so far: As ...
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0answers
22 views

GF(2^n) Multiplication using normal base

I want to implement the multiplication using normal basis in the binary field $GF(2^n)$ (where n=163, 233 for example). The multiplication with normal basis is performed by $c_0=A*M*B^T$, where $M$ is ...
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1answer
31 views

Possible ranks of a $n!\times n$ matrix with permuted rows

Let $a_1,\cdots,a_n$ be $n$ arbitrary real numbers. Form the $n! \times n$ matrix $M$ whose rows are obtained by permuting the $n$ numbers given. Find all the possible ranks of such a matrix. ...
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2answers
69 views

Is this notation on the restriction of a function in group theory common?

If $f: X \rightarrow Y$ is a function between sets $X$ and $Y$, then a common notation to use when we want to restrict $f$ to a certain domain $X' \subset X$ is $f|_{X'}: X' \rightarrow Y$. I'm doing ...
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0answers
25 views

Efficient algorithm for solving a bilinear Diophantine system

I wonder if there is an efficient algorithm for finding an integer solution $(x_1, \dots, x_n)$, $(y_1, \dots, y_n)$ for the following type of system of equations $$ a_{1,1} x_1 y_1 + a_{1,2} x_1 y_2 ...
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1answer
66 views

Representation of $Q_8$ over $\mathbb{R}$

I'm trying to solve the following problem, Give an example of a finite group $G$ and its irreducible representation $L$ over $\mathbb{R}$ such that the division algebra $Hom_G(L, L)$ is isomorphic ...
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1answer
39 views

Show that these are the only $\mathbb{R}[x]$-submodule of $\mathbb{R}^2$ [on hold]

Let $T$ be the linear mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$ that is given by a clockwise rotation of $\frac{\pi}{2}$. $T$ makes $\mathbb{R}^2$ an $\mathbb{R}[x]$-module. I want to show ...
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1answer
29 views

If $ p\neq q$ are odd prime integers then $(\mathbb{Z}/ pq\mathbb{ Z})^*$ is not cyclic

This is a question from Aluffi's Algebra Chapter 0, which I am self studying. Specifically this is from Chapter 2, Page 69, Question 4.10 Let $p\neq q$ be odd prime integers; show that $(\mathbb{Z}/ ...
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1answer
30 views

Prove that a commutative ring without proper ideals is a field [duplicate]

Let $R$ is a commutative ring which has no proper ideals. Prove that $R$ is a field.
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1answer
42 views

Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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2answers
31 views

Before proving that $Aut(G)$ is a subgroup of $S_G$ …

Exercise 37 Ch. 9 of the book Abstract Algebra by T Judson : We will denote the set of all automorphisms of G by $Aut(G)$. Prove that $Aut(G)$ is a subgroup of $S_G$ , the group of permutations ...
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1answer
74 views

some confusions about the concepts of algebra

Recently I tried to learn Algebra(Revised third edition) with the book written by Serge Lang. Since I have not covered all topics in the elegant book but now just view it as a reference for some ...
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1answer
66 views

Matrices and prime numbers

Let $ p $ be a prime number and \begin{align} K=\left\{ \begin{pmatrix} a &b \\ c& d \end{pmatrix} \mid a,b,c,d \in \left\{0,1,\ldots,p-1 \right\}, \right. & a+d \equiv 1 \!\!\!\! ...
6
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1answer
75 views

On the converse of Schur's Lemma

Let $G$ be a finite group and $F$ a field with $\mathrm{char}(F)=0$ or coprime to $|G|$. Let $V$ be a $FG$-module in a way that every $ FG$-homomorphism $ f : V \to V $ is given by $f(x)= \lambda x ...
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5answers
59 views

Proving that the groups $S_3$ and $D_6$ are isomorphic [duplicate]

In particular, $S_3$ is the group of permutations of $\{1,2,3\}$, and $D_6$ is the dihedral group of symmetries of the triangle (written as $D_{2\cdot 3}$). In generator-relation form, $D_6 = ...
3
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1answer
48 views

Free group uniqueness from the universal property

Just a note, I'm using Dummit and Foote (specificially second edition Prentice Hall, section 6.3 "A word on free groups") as my reference material. I also found a similar question here, but this is ...
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1answer
20 views

Trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$

I am trying to calculate $\operatorname{dim}H_1(RP^2$#$T^2;Q)$ and $\operatorname{dim}H_1(RP^2$#$T^2;F_2)$ I know that $RP^2$#$T^2$~$RP^2$#$K^2$ and that $X(M$#$N)$=$X(M)+X(N) -2$ where X is the ...
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0answers
20 views

Binary operation on Graphs

Are there "binary operations" on graphs, which make the set of all graphs, a commutative ring or a field For example $G_1 \cdot ( G_2 + G_3) = G_1 \cdot G_2 + G_1 \cdot G_3$ By a graph I mean ...
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1answer
35 views

For each polynomial $p \in K[t]$, there exists another polynomial $g$ such that $\{p(\epsilon_i): 1\leq i \leq n\}$ is its set of roots

I'd like to solve the following problem: Let $K$ be a field, $f \in K[t]$ a polynomial of degree $n$ and $E$ a splitting field of $f$ over $K$ in which $f$ has $n$ distinct roots ...
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2answers
31 views

The product $J^{n}A$ of ideals

Suppose that $A$ and $J$ are two ideals of a ring $R$. I can't understand the following implication: If $JA = A$ then $J^{n}A = A$ for all $n > 0$. True that $J^{n}A$ is an ideal of $A$ for all ...
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1answer
51 views

Prove there isn't a non-zero nilpotent element of a finite ring

Let $(A, +, \cdot)$ be a finite ring with $n$ elements having the property: $x^n \ne1, \forall x \ne 1, x \in A$. Prove $0$ is the only nilpotent element of $A$. My attempt Suppose there is ...
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2answers
21 views

problem about inner semidirect product

Let $G$ be a group which is the product $G=NH$ of subgroup $N,H\subset G$ where $N$ is normal. Let $N\cap H=\{1\}$. I am trying to show that that there is an iso $G\cong N\rtimes H$, with the ...
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0answers
16 views

Grobner basis and number theory [closed]

Could someone help me? What is the relationship between grobner basis and number theory?
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1answer
75 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
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3answers
22 views

To show that Z(G) = $\cap_{a \in G} C(a)$

To show that Z(G) = $\cap_{a \in G} C(a)$ (Intersection of all subgroups of form C(a)) Let $a \in Z(G)$. Then $ax=xa$ for all $x$ in G. IN particular we can say that $ax_1=x_1a$ and $ax_2=x_2a$ and ...
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1answer
50 views

Problem about subgroup of $D_n$

Prove that every subgroup of $D_n$ , either every member of subgroup is a rotation or exactly half of them are rotations. Intuitively, if every member is a rotation then they will form a subgroup ...
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1answer
34 views

When will the classification of a property of a group be complete

I am studying pronormal Hall $\pi$-subgroups of a finite group $G$ All Hall $\pi$-subgroups of a finite solvable groups are known to be pronormal as this follows from Hall's theorem Recently, it has ...
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2answers
32 views

How to prove isomorphic? [duplicate]

(Q*,•) and (R*,•) is isomorphic. This is false. Why this problem false?? Would you ask to me counter-example?
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0answers
23 views

Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$ [duplicate]

Suppose $n$ is even positive integer and $H$ is a subgroup of $\mathbb Z_n$. Prove that either every member of $H$ is even or exactly half of members of $H$ are even. Same question have been asked ...
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35 views

Show that in a group order of element is less than or equal to order of group [closed]

Show that in a group order of element is less than or equal to order of group. This is a Question from Gallian. Please provide hints on how to start this? THanks
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2answers
57 views

Prove that if $G$ is cyclic and infinite then $G$ is isomorphic to $\mathbb{Z}$

Assume $G$ is generated by $a$, so $G = \langle a\rangle$. Since $G$ is infinite for all $m \in \mathbb{Z}$, $a^m \neq e$. Suppose $a^h = a^k$ then $a^h\cdot a^{-k} = a^{h-k} = e$, but this is a ...
1
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0answers
42 views

The software for finite field arithmetic

Is there any software, library,or toolkit that support arithmetic with normal basis on $GF(2n)$ field? What is the best one? Especially, which software can implement the conversion between normal ...