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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$R$ 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 $r$ ...
2
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1answer
55 views

What makes it legitimate to multiply both sides?

Having the proof of the cancelation law for multiplication: $$cb=ab$$ $$(cb)b^{-1}=(ab)b^{-1}\tag{Inverse}$$ $$cbb^{-1}=abb^{-1}\tag{Associativity}$$ $$c\cdot 1=a\cdot 1\tag{Indentity}$$ $$c=a$$ ...
2
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2answers
65 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
34 views

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

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 $0$?
4
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1answer
41 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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0answers
10 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
23 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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0answers
11 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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0answers
36 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 ...
0
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1answer
19 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 ...
5
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2answers
102 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
23 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 ...
0
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0answers
16 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 ...
2
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1answer
19 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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0answers
30 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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0answers
15 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
48 views

Very easy question of ring theory

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 ...
1
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1answer
24 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 ...
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1answer
21 views

Finding $|E|$, where $E$ is the Splitting Field of $x^8-1$ over a Field of $4$ Elements.

This is my attempt to find $\vert E \vert$, which is the order of the field $E$. If I am on the wrong track, please guide me to a technique that will work with more general fields and polynomials. We ...
2
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3answers
54 views

Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$

Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$ My thoughts: $|a|=|b|=2\implies a^2=e$ and $b^2=e$ I see that the group cannot be abelian as the order wont be ...
0
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1answer
20 views

$ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $, $ G/N_{1}\cap N_{2} \cong G $?

Let $ G $ is a finite group and $ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $ that $ N_{1} \neq N_{2} $. Suppose $ G/N_{1} $ and $ G/N_{2} $ are supersolvable. Then $ G $ is ...
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0answers
66 views

What are super-translations?

There's been a lot of news lately about a possible solution to the black hole information paradox from a presentation given by Stephen Hawking to the KTH Royal Institute of Technology in Stockholm. ...
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2answers
38 views

How can I prove that $g\cdot H \cdot g^{-1}$ is also finite and has the same number of elements that $H$?

Suppose $G$ a group and $H$ is a finite subgroup of $G$ also $\forall g \in G$ the set $g\cdot H \cdot g^{-1}=\{ g\cdot h \cdot g^{-1} : h\in H\}$, is a subgroup of $G$. Prove that $g\cdot H ...
2
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1answer
66 views

How to prove that G is a cyclic Group? [duplicate]

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
0
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1answer
51 views

Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible [duplicate]

Let $p$ a prime number. Show that the polynomial $X^p-X-1 \in F_{p}[X]$ is irreducible using this hint: If $a$ is a root of $X^p-X-1$, show that $a^{p^{p}}=a$, who is the extension ...
0
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1answer
28 views

The sum of an isogeny and its dual for the Frobenius homeomorphism

This is from page 150 of Silverman's "The Arithmetic of Elliptic Curves". Any my only questions is: How you can conclude that $[a]=\phi+\hat{\phi}$? I tried to use the formula on page 85 which ...
4
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1answer
23 views

Diameter of unitary group.

Define a function$$N: \text{End}_\mathbb{C} \to \mathbb{R}_{\ge 0},\text{ }N(a) := \max_{\{v \in V\,:\, |v| = 1\}} |av|.$$ What is $$\max_{a, b \in U(V)} N(a - b),$$the "diameter" of the group ...
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0answers
21 views

$ G $ is soluble, then every properties of $ G $ is inherited by $ G/N $ ? [on hold]

Let $ G $ be a finite group and $ G $ is soluble. Suppose $ N $ be a normal minimal subgroup of $ G $. Then every properties of $ G $ is inherited by $ G/N $ ?
18
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3answers
399 views

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: ...
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1answer
101 views

Prove $R$ is a finite ring [on hold]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
2
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0answers
21 views

Border rank of tensors

Can anyone help me find the rank and border rank of the following tensor: \begin{align} T=a_{11}\otimes b_{11}\otimes c_{11}+a_{12}\otimes b_{21}\otimes c_{11}+a_{11}\otimes b_{12}\otimes c_{12}\\ ...
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0answers
33 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
0
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0answers
27 views

what is the minimal condition for two elements to create same field extension?

Given a field $K\subset E$, with $\alpha,\beta\in E$, such that $K(\alpha)=K(\beta)$. What can we then say about $\alpha$ and $\beta$? If the extension is finite, then $\alpha$ is a linear ...
2
votes
1answer
29 views

$U(\mathbb{C}^n)$, $SU(\mathbb{C}^n)$ connected subsets of $M_n(\mathbb{C})$?

As the title suggests, is $U(\mathbb{C}^n)$ a connected subset of $M_n(\mathbb{C})$? How about $SU(\mathbb{C}^n)$?
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19 views

Strassen's Laser Method Technique AND Tensors in matrix multiplication algorithms

I understand the first algorithm presented by Strassen in 1968, for fast matrix multiplication. This was the first improvement to the naive approach of multiplying matrices. Thereafter, he went on to ...
4
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0answers
30 views

Irreducibility of $p(x)$ implies that of $p(x+c)$ only when taken over a field?

$R$ is a ring and $R[x]$ is the polynomial ring over $R$ . $c$ is any fixed element of $R$ . Then the map $f(x)\mapsto f(x+c)$ is an isomorphism from $R[x]$ to itself. Now ...
0
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1answer
14 views

Proving that an adjacency transposition is the product of odd number of adjacencies.

A transposition in $S_n$ of the form $(i \ i + 1)$ is called an adjacency. I am trying to prove that, Given $i ∈ \{1, . . . , n − 1\}$, if $i < j$, the transposition $(i \ j)$ is a product of an ...
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0answers
35 views

Find $\operatorname{min}(\omega\sqrt{2},\mathbb{Q})$ [duplicate]

Let $\omega$ be a primitive third root of unity with $K=\mathbb{Q}(\omega,\sqrt{2})$. Find $\operatorname{min}(\omega\sqrt{2},\mathbb{Q})$ Could anyone tell me how to find this? and generally which ...
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0answers
39 views

I need the paper “ J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365” [on hold]

I need this paper " J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365", But i not found. Who can help me to find this paper ?
0
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1answer
30 views

Irreducibility of a polynomial

Given that $\mathbb F$ is a field and $\mathbb F[x]$ is the polynomial ring over $\mathbb F$. $\ \ $If the polynomial $a_{0}+a_{1}x+a_{2}x^{2}+......a_{n}x^{n}$ is irreducible over ...
3
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2answers
56 views

Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $\mathbb Z$-module for $m < n$?

Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $ \mathbb Z$-module for $m < n$ ? I think it's not possible. It might be a easy problem for some of you, but I really don't ...
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1answer
36 views

an example of a normal group that is not abelian

Can anyone please tell me an example of a normal group that is not abelian? I read that in a normal group every left coset is equal to right coset.
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2answers
34 views

finite vs infinite set function composition

If there is a set $X$ which is finite with $f : X \rightarrow X$ and $g: X \rightarrow X$, then $f \circ g = 1_X$ iff $g \circ f = 1_X$. How is it true for finite sets? I'm not too sure, but the ...
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1answer
35 views

An example to prove that all cosets do not form a group

I read that only cosets (G/H) such that H is a normal subgroup form a factor group. Can anyone tell me an example of a case where cosets do not form a group ?
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3answers
64 views

Find the subgroup of $GL(2,\mathbb{C})$ generated by two matrices $A$ and $B$.

Find the subgroup of $GL(2,\mathbb{C})$ generated by the matrices $A$ and $B$, where $A=\begin{pmatrix} 1 & 0\\ 0 & i \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1\\ -1 & 0 ...
0
votes
1answer
40 views

How to find all homomorphism $\delta :V_4 \to \mathbb{C}^{*}$.

How to find all homomorphism $\delta :V_4 \to \mathbb{C}^{*}$. Where $V_4$ is Kleins 4 group and $\mathbb{C}^{*}$ is multiplicative group of nonzero complex numbers.
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34 views

What is the relation between $\mathcal{G}_{12}, \mathcal{G}_{23}, \mathcal{G}_{13}$?

please help me to find this. Suppose that the number of group homomorphism from the group $G_1$ to $G_2$ is $\mathcal{G}_{12}$, from $G_2$ to $G_3$ it is $\mathcal{G}_{23}$ and from $G_1$ to $G_3$ ...
3
votes
1answer
15 views

Induced coaction on a vector space.

Let $H$ be a Hopf algebra and let $V$ be an $H$-module. Then we have an action $H \times V \to V$ given by $(h, v) \mapsto h.v$. This action induces an $H$-action on the dual vector space $V^*$ of $V$ ...
2
votes
1answer
54 views

Intuition for Burnside's Lemma (aka Cauchy-Frobenius Lemma)

Here is the theorem: Lemma: Let a group $G$ act on a set $S$. Define $\text{Fix}(g)$ as the set of all elements in $S$ fixed by $g$ under this group action. Then the number of distinct orbits of ...