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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$ 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, ...
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What means group-theoritic property $ P $?

Suppose that $ \vert G \vert $ is minimal subject to the condition that permutizers of maximal subgroups of $ G $ are equal to $ G $, $ G $ is a nonsupersolvable group, and $ G $ has a group-theoritic ...
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20 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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13 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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40 views

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

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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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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17 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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21 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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24 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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31 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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14 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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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
17 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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51 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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35 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
38 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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29 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
26 views

Dimension of algebra over $\mathbb{C}$.

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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27 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 ...
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29 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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54 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 of numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
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$\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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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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16 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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27 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 $d(G)=d(\frac{G}{\Omega_1(Z(G))})$ ...
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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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39 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 ...
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37 views

Is $\langle\mathbb Q^+, *\rangle$ a monoid?

Q: Given the set of positive rational numbers $\mathbb Q^+$, the operation is multiplication$~*$. Is $\left<\mathbb Q^+, *\right>$ a monoid? My answer is: $ \forall x, y, z \in \mathbb Q^+$, ...
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1answer
24 views

$m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?

As the title suggests, how do I see that an $m \times n$ matrix gives rise to a well-defined map from $\mathbb{R}^n$ to $\mathbb{R}^m$?
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1answer
31 views

product of subgroups and group G

Is there any example of two subgroups H and K of G whose product give G i.e. G = HK but none of which is normal in G
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Finite Modules Isomorphism

For a general ring $R$ i think that is not true that 2 R-Modules are isomorphic even if they have the same cardinality, as in the case of vector spaces but i would like to have some example of that ...
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1answer
30 views

$R$ is a ring with identity. Why from $f(1)=0$ it's concluded that $\forall r\in R; f(r)=0$?

The original question is this: Let $R$ be a ring with identity and $\mathbb{C}$ the ring of complex numbers. Suppose $f,g:R\rightarrow \mathbb{C}$ are two ring homomorphisms such that for every $r$ ...
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144 views

What makes it legitimate to multiply both sides?

Having the proof of the cancelation law for multiplication: $$cb=ab$$ $$(cb)b^{-1}=(ab)b^{-1}\tag{Inverse}$$ $$cbb^{-1}=abb^{-1}\tag{Associativity}$$ $$c\cdot 1=a\cdot 1\tag{Indentity}$$ $$c=a$$ ...
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175 views

Prove that G is a group

The exercise is: Let $g\in G$. $G$ is a group. Prove that $G=\{gx:x\in G$}. I know the the definition of group but the proof that is in the book is the next one: Let $H=\{gx:x\in G\}$ ...
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1answer
38 views

$F$ is subfield of complex field $\mathbb{C}$. Show that $F$ is field with characteristic $0$ [on hold]

We know that subfield of field is set $F$ of complex number which itself is a field under usual multiplication and addition. but how to show that it has characteristic $0$?
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1answer
48 views

Prove that : $n \mid \varphi (a^{n}-1)$ $a>1$

Prove that : $n \mid \varphi (a^{n}-1)$ $a,n$ positive integers wih $a>1$ I know that $a$ has multiplicative order $n$ in the ring of integers modulo $a^{n}−1$ and the order of the group of ...
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16 views

Action of the Symplectic Group on Siegel Upper Half Plane

Given $G= \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in Sp_{2n}(\mathbb{R})$ one can define an action on the symmetric $n \times n$ complex matrices with positive definite imaginary part by ...
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1answer
28 views

Order of Automorphis group [on hold]

Let $ G $ is a finite solvable group and $ N $ be a normal minimal subgroup of $ G $ that $ G = MN $ for maximal subgroup $ M $ of $ G $, which $ M \cap N = 1 $. Let $ \vert N \vert = 4 $. Then $ ...
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21 views

looking for example of infinite p-group of nilpotency class 2

Is there infinite p-group of nilpotency class 2? If p=2 or p=3 would be better. and I prefer simple examples
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87 views

Good algebra book to cover these topics?

I will be studying two algebra modules next year and I am looking for a comprehensive book that will cover both of them, however due to having very minmal exprience with algebra I am looking for your ...
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20 views

field extension $F\subset E$ with both separable and inseparable elements

Can someone please give me an example of a field extension $F\subset E$ such that E\F has both separable and inseparable elements? if $F(\alpha)$ is a simple extension of F, and if $\alpha$ is ...
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124 views

The ring $\mathbb{C}[x,y]/\langle xy \rangle$

What can be said about the ring $\mathbb{C}[x,y]/\langle xy \rangle$? I was very certain that $$\mathbb{C}[x,y]/\langle xy \rangle \cong\mathbb C[x] \oplus\mathbb C[y]$$ since the elements in ...
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2answers
27 views

Fibers of an Element

Prepping myself for a graduate abstract course and we are using Dummit and Foote's Text. We are starting on Chapter 1, so I thought it would be a good idea to go over chapter 0. There is a term that ...
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1answer
25 views

Inner automorphisms Inn(D4).

We need to show that elements of $Inn(D_4)$ are distinct , where , $Inn(D_4)= \phi_{{R_0}} , \phi_{{R_{90}}} , \phi_{H} , \phi_{D}$. Is it sufficient to construct a Cayley table for the elements of ...
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1answer
20 views

invariant properties between p-group and it's automorphism

Let $G$ be a p-group and $Aut(G)$ be group of automorphisms of $G$ which properties of $G$ can help us with studding $Aut(G)$? for example If $G$ is infinite/finite does this guaranty $Aut(G)$ be ...
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36 views

How to construct a ring $ R$ such that $(Spec(R), \tau)$ is not a Sequential Space where $\tau$ is the Zariski Topology on $R$

How to construct a ring $ R$ such that $(Spec(R), \tau)$ is not a Sequential Space where $\tau$ is the Zariski Topology on Spec(R). I've just learned about Zariski topology so I really don't have ...
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17 views

Finite ring having $2^n-1$ invertible elements [duplicate]

Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ ...
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1answer
49 views

Very easy question of ring theory [on hold]

Can we introduce $R/I$, where $R$ is a ring and $I$ is a sub-ring of $R$? Thanks a lot.
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2answers
37 views

Field and Field Axioms.

I wanted to ask what are field and field axioms? I have tried looking on Wikipedia and Wolfram But They are too are advanced and I cant a understand one bit.So please any help would be much ...
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1answer
27 views

Endomorphism structure of the Klein four-group

I am reading the Algebra by Grillet, this is ex 17(-18), pag. 22. I understand that $V$ can be viewed as a two dimensional vector space over $\mathbb{F}_2$. Noticed this, it is easy to see that ...