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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Elements that aren't left zero divisors are invertible for certain group algebra

Let $G$ be a finite group and $F$ a finite field with co-prime orders. Show that in the group algebra $F[G]$, if $x \in F[G]$ is not a left zero divisor then it is invertible. Thoughts so far: By ...
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8 views

There exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$.

$I\trianglelefteq \Bbb F[x]$. I want to prove that there exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$. I guess this means that I am meant to show that we have closure from the ring ...
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25 views

Elementry of analysis [on hold]

I don't understand this question, could you please define the question for me and tell me what should I do? Consider a particular device capable of arithmetic computation. Consider the operation ...
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33 views

The number of all groups with n elements? [on hold]

Let $n$ be a positive integer number. How many groups of $n$ elements (which are not isomorphic) ?
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1answer
83 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
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2answers
37 views

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$? In my title, I should be clear that the additive inverse should be unique. But how does this help? I dont even get why ...
-1
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0answers
32 views

Left remainder when dividing by $x-b$

Give a polynomial $p(x) = a_0 + a_1 x + ... a_n x^n \in \mathcal R[x]$ ($\mathcal R $ is any ring with unity), the book says when dividing $p(x)$ by $x-b \quad (b\in \mathcal R)$, the left remainder ...
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1answer
38 views

For an ideal $I \subset \mathbb{C} [x_1, … , x_n]$ show an iff about finiteness

For an ideal $I \subset \mathbb{C} [x_1, ... , x_n]$ show that dim$_{\mathbb{C}}R/I$ is finite iff $I$ is contained in only finitely many maximal ideals. Thoughts so far: I'm not sure how to get ...
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21 views

Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $.

Let $ G $ is a soluble group and $ N $ minimal normal subgroup of $ G $. Let $ F = Fit(G) $. Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $. I show ...
2
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1answer
27 views

Linear algebra: generalize from characteristic $0$ a problem about polynomial coefficients.

Let $K$ be a field, and let $F$ be a subfield of $K$. Assume that $F$ is infinite. Let $p(x)$ be a polynomial in one variable with coefficients in $K$, and suppose that $p(a) \in F$ whenever $a \in ...
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1answer
18 views

Linear transformations for fixing the line $y = 0$

The professor says that the subgroup for "stabilizing" the line $y = 0$ is $$A = \begin{bmatrix} a & c \\ 0 & d \end{bmatrix}$$ because in order to fix the first basis vector, $b = ...
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9 views

On Weisner Method

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
5
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1answer
30 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
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1answer
12 views

prove unique minimal normal subgroup of soluble group by $ P_{G}(M) > M $

Let $ G $ be a soluble group. If $ P_{G}(M) = \langle y\in G | \langle y \rangle M = M\langle y \rangle \rangle > M $for any subgroup $ M $ of prime power index in $ G $, then every chief factor ...
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8 views

$\mathbb{C}\{X\}^\chi$ a $G$-stable subspace of $\mathbb{C}\{X\}$?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = SL_2(\mathbb{F})$. The group $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group ...
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2answers
31 views

$(a,b) \mathbin\# (c,d)=(a+c,b+d)$ and $(a,b) \mathbin\&(c,d)=(ac-bd(r^2+s^2), ad+bc+2rbd)$. Multiplicative inverse?

Let $r\in \mathbb{R}$ and let $0\neq s \in \mathbb{R}$. Define operations $\#$ and $\&$ on $\mathbb{R}$ x $\mathbb{R}$ by $(a,b) \mathbin\#(c,d)=(a+c,b+d)$ and $(a,b) ...
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44 views

A quick Galois Theory question

Let $\mathbb{F}_q$ be a finite field of order $q$ where $q$ is a prime power. For any $d \in \mathbb{N},$ we have an inclusion $\mathbb{F}_{q^d} \subseteq \overline{\mathbb{F}}_q.$ Both ...
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44 views

Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
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57 views

Number of matrices $A \in M_n(\mathbb{F}_q)$ where $A^2 = 0$.

What is the number of matrices $A \in M_n(\mathbb{F}_q)$ for which $A^2 = 0$ (as a function of $n$ and $q$)?
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1answer
56 views

Is there a “unique factorization theorem” for finite groups?

Sometimes it is difficult for me to understand what a group seems like. For example, the dihedral group $D_5$ is easy to visualise when I think it of as a "product" of two cyclic groups $C_2$ and ...
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4answers
44 views

Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain

Here is an exercise(p.129, ex.1.15) from Algebra: Chapter 0 by P.Aluffi. Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. The implication part makes no problems, ...
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2answers
37 views

Is $V$ a simple $\text{End}_kV$-module?

Let $V$ be a finite-dimensional vector space over $k$ and $A = \text{End}_k V$. Is $V$ a simple $A$-module?
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23 views

Injective hull of a simple module

Let $M$ be an indecomposable injective right module over a right Artinian ring $R$, so $M$ has exactly one associated prime ideal $P$ (Lectures on Modules and Rings, T.Y. Lam). Now, $R/P$ is a simple ...
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54 views

Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable. I can see that the polynomial $x+y$ is in ...
3
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1answer
34 views

$ S_{4} $ has a sylow tower?

A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, ...
3
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1answer
27 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
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27 views

Largest algebra in a vector space

Let $V$ be a vector space and let $C$ be a subset of $V$ that is closed under a bilinear operator $\langle \cdot, \cdot \rangle: V \times V \rightarrow V$. Let $A \subset V$ be an algebra containing ...
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50 views

Irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$ [on hold]

We wish to find all the irreducible polynomials over the field $\mathbb{Z}_3[x]/\left\langle x^2+1\right\rangle$. I came across this problem in my course on advanced algebra. I have little knowledge ...
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1answer
18 views

Determining a lower bound on the order of a group based on its presentation

I am reading Abstract Algebra book by Dummit and Foote (3-rd edition). On pages 26-27 they define a dihedral group: $D_{2n} = \langle r,s | r^n = s^2 = 1, rs = sr^{-1} \rangle$ The authors ...
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0answers
26 views

Quiver algebra as a wreath product?

I'm having trouble understanding a definition of a quiver Hecke algebra. Suppose $k$ is a commutative ring, and $\Omega$ a finite set. We build a quiver $Q_{\Omega,n}$ with vertex set $\Omega^n$. ...
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30 views

Show that for each $\alpha \in \mathbb{F}_{q}$ there are $q^{m-1}$ elements $\beta$ in $\mathbb{F}_{q^{m}}$ such that $Tr(\beta) = \alpha$

Show that for each $\alpha \in \mathbb{F}_{q}$ there are $q^{m-1}$ elements $\beta$ in $\mathbb{F}_{q^{m}}$ such that $Tr(\beta) = \alpha$ I am trying to solve this question which looks like to be ...
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0answers
28 views

Commutative rings and zero divisors [on hold]

If the ring R is commutative and a belongs to R is not a zero divisor, show that the principal ideal of all multiples of a is a free R-module on one generator.
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35 views

Definition of an exponential polynomial

From Wikipedia: For formal exponential polynomials over a field $K$ we proceed as follows. Let $W$ be a finitely generated $Z$-submodule of $K$ and consider finite sums of the form ...
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17 views

An equality for commuting endomorphisms

Suppose $f,g$ are injective endomorphism of abelian group $M$ and $f\circ g=g\circ f$,is it true that $$\left(f^{-1}(g(M))+g(M)\right)/g(M)\cong \left(g^{-1}(f(M))+f(M)\right)/f(M)?$$
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2answers
21 views

Why is the arrow ideal $R_Q$ of a finite, connected, acyclic quiver $Q$ equal to the Jacobson radical?

If $Q$ is a finite, connected, acyclic quiver, why does the arrow ideal $R_Q$ equal the Jacboson radical $J$ of the quiver algebra $k(Q)$? It comes up in showing that the quotient $k(Q)/R_Q$ is a ...
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1answer
24 views

Can the order on an ordered, cancellative monoid be extended to its Grothendieck group?

Suppose we have an ordered, cancellative monoid and we wish to apply the Grothendieck group construction to it. Can the total order be extended to the larger group? Example: consider the ordered ...
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1answer
57 views

I need this book by Michael Weinstein, Between nilpotent and solvable [on hold]

I need this book by Michael Weinstein, Between nilpotent and solvable. I live in iran and i can't find this book. Please help me.
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40 views

$ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $.

Let $ G $ be a soluble group, $ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ for prime $ p $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $. For proof i employ the ...
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2answers
40 views

about a maximal normal subgroup of a $p$ group.

i'm studying bhattacharya's basic algebra. it introduces the concept of the group action in chapter 4 and proves the class equation. and derives simple properties of $p$ group using the equation. the ...
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1answer
31 views

Finding all homomorphisms [duplicate]

How can I solve this question Find all group homomorphisms from $\mathbb{Z_{5}}$ into $\mathbb{Z_{12}^{x}}$?
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1answer
29 views

Dimension of algebra over $\mathbb{C}$. [on hold]

Let $n \ge 2$, and let $A$ be the (unital and asociative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is the ...
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2answers
39 views

Finding ring isomorphisms

Let $A$ be a ring with $0\neq 1$ such that $x^4=1, \forall x\in A$, with $x\neq 0$. My question is: to which ring is $A$ isomorphic? $A$ can be, for example, isomorphic to $\mathbb{Z}_2$. The ...
2
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1answer
39 views

A question about finitely generated projective modules

Let $A$ be a commutative ring with unity and let $P$ be a finitely generated projective $A$-module. For $any$ $A$-module $M$, how does one show that $\operatorname{Hom}_A(P,A) \otimes_A M \simeq ...
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2answers
65 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
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3answers
62 views

$\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has form $X \mapsto AXA^{-1}$.

As the title suggests, what is the easiest way to see that any $\mathbb{C}$-algebra automorphism of $M_n(\mathbb{C})$ has the form $X \mapsto AXA^{-1}$ for some fixed $A \in GL_n(\mathbb{C})$?
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1answer
15 views

$ G $ is $ p $-supersolvable group . $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Show $ Q \unlhd G $.

Let $ G $ is a finite $ p $-supersolvable group for odd prime numbers . Suppose $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Now i'll show $ Q \unlhd G $. Since $ G $ is $ p $-supersolvable, then $ ...
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2answers
17 views

Ring homomorphism from field

If we have homomorphism from field K to ring R, does that mean that we have ring homomorphism but K is a field? I have trouble understanding this. Thank You very much for your help.
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1answer
36 views

If G is finite p-group then $d(G)=d(\frac{G}{\Omega_1(Z(G))})$

Let $G$ be a finite p-group such that $G$ has no non-inner automorphism of order p leaving Φ(G) elementwise fixed If $\Omega_1(Z(G))\le G'\le \Phi(G)$ how we can get $d(G)=d(\frac{G}{\Omega_1(Z(G))})$ ...
2
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0answers
21 views

Tangent to dual curve.

Let $C$ be a smooth projective plane curve, let $P \in C(k)$, and let $\ell$ denote the tangent line to $C$ at $P$. Let $C^*$ denote the dual curve to $C$, in the dual plane $(\mathbb{P}^2)^*$ (the ...
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2answers
41 views

Intuition for a ring homomorphism?

A map $f: A \to B$ between two rings $A$ and $B$, is called a ring homomorphism if $f(1_A) = 1_B$, and one has $f(a_1 + a_2) = f(a_1) + f(a_2)$ and $f(a_1 \cdot a_2) = f(a_1) \cdot f(a_2)$, for any ...