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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3
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1answer
22 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 ...
0
votes
0answers
3 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 ...
-1
votes
1answer
18 views

Order of Automorphis group

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 $ ...
0
votes
0answers
8 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
0
votes
0answers
31 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
votes
1answer
18 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
votes
2answers
75 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 ...
1
vote
2answers
22 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
15 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
votes
1answer
17 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 ...
1
vote
0answers
25 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 ...
1
vote
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 $2^n-1$ ...
-1
votes
1answer
45 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.
-1
votes
2answers
35 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
vote
1answer
22 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 ...
0
votes
1answer
20 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
votes
3answers
44 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
votes
1answer
19 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 ...
4
votes
1answer
59 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. ...
0
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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
votes
1answer
65 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
votes
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 ...
4
votes
1answer
22 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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votes
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
votes
3answers
390 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: ...
-6
votes
2answers
97 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
votes
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}\\ ...
7
votes
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
votes
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 $SU(\mathbb{C}^n)$?
0
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0answers
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
votes
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
votes
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 ...
2
votes
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 ...
1
vote
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
votes
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
votes
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 ...
0
votes
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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vote
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 ...
0
votes
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 ?
0
votes
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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0answers
33 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 $G_3$ ...
2
votes
1answer
14 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 ...
0
votes
1answer
53 views

Let $F$ be the set of all functions of the form $f: t\to \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$. Is $F$ an integral domain? Is it a field?

Let $F$ be the set of all functions $f$ from $\mathbb{R}$ to itself of the form $$f: t\to \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$$ where $a_k$ and $b_k$ are real numbers and $n$ is a natural ...
-1
votes
0answers
52 views

Ideals in Algebra [on hold]

Is there any geometric interpretation of ideals in algebra (like for instance one can intuitively imagine co-sets in terms of affine subspaces of a vector space)? Are there any instances in fields ...
9
votes
3answers
57 views

Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

What is the smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi-direct product of cyclic groups? So finite abelian groups are ruled ...
4
votes
1answer
48 views

$O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$

During a lecture of a Lie Algebras yesterday, the professor of the class stated the following fact without proof $O_{2n}(\mathbb{R}) \cap GL_{n}(\mathbb{C})=U(n)$ Note that we are viewing ...
0
votes
4answers
40 views

Prove by definition that $(x,2)\subset\mathbb Z[x]$ is a maximal ideal

When the polynomial ring $\mathbb{Z}[x]$ is quotiented by the ideal $(2,x)$ we get a field as $\mathbb{Z}[x]/(x,2)\cong\mathbb{Z}/(2)\cong\mathbb{Z}_{2}$ which is a field. But I ...