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Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

-1
votes
1answer
38 views

Set Theory, Infinite chain of subsets [on hold]

Given an infinite chain of subsets $S_1\subseteq S_2 \subseteq S_3...$. Consider a subset $N\subset\bigcup\limits_{i=1}^{\infty} S_{i}$, can we say that $N\subseteq S_i$ for some $i$. If so, how do we ...
3
votes
1answer
40 views

Size of conjugacy class in subgroup compared to size of conjugacy class in group

Given: $\bullet$ A finite group $G$, an index 2 subgroup $H$, an element $a \in H$ $\bullet$ $[a]_H$ and $[a]_G$ are the conjugacy class in $H$ of $a$ and the conjugacy class in $G$ of $a$, ...
1
vote
0answers
14 views

Prove: A nonempty subset B of a ring A is closed with respect to addition and negatives iff B is closed with respect to subtraction [duplicate]

Can I say something like, ∀ x,y ∈ B, x + (-y) ∈ B ⇒ x - y ∈ B by definition of subtraction?? I'm lost
1
vote
1answer
22 views

Group as direct sum of cyclic groups

What are necessary conditions for a cyclic group $G$ to be a direct sum of cyclic groups? I saw somewhere that $G$ must be a non $p$-group. But I couldn't prove it. Thank you for your hints/help
0
votes
1answer
50 views

determine the cardinality and all inverrtible elements of $\mathbb{Z}[x]/(4,x^2)$.

I've got to determine the cardinality and all inverrtible elements of $\mathbb{Z}[x]/(4,x^2)$. I know that in this quotient ring $4=0$ and $x^2=0$. So $x^2=4$. The only elements would be $\{1,2,3, x,...
0
votes
0answers
40 views

A finite group of order $n$ has exponent $n$.

Definitions: The order $|G|$ of a group $G$ is the number of elements of $G$. The exponent of a group is an integer $n$ such that $x^n = e$ for all $x\in G$ ($e$ is the neutral element). ...
1
vote
1answer
30 views

Why is the number of Sylow 2 subgroups of simple group with order 60 not able to be 1 or 3?

I want to show that a simple group of order 60 is isomorphic to $A_5$. In the process, I am stuck at the part in which I have to show that the number of Sylow 2 subgroups (whose orders are 4) cannot ...
1
vote
1answer
22 views

Finding all composition series of $S_3 \times Z_2$

It was easy to find the composition factors of $S_3 \times Z_2$. However, I cannot see how to find 'all' possible composition series of this group. It seems like a formidable work to me... Could ...
0
votes
0answers
22 views

Mathematical Foundations of Quantitative Finance [closed]

I really need help. I want to study an MSc in Finance but my maths has never been that good. I have got a 2:1 in business accounting but this is a new level of maths and would be grateful for some ...
1
vote
1answer
29 views

Coprime numbers and intersection of subgroups

We know that the intersection of two subgroups is {identity element} if their orders are coprime (proof via Lagrange's Theorem). But can we say: if the intersection of two finite subgroups is {...
0
votes
0answers
34 views

Application of Maschke's Theorem to $S_4$

I'm trying to find something out about an algebraically closed field $k$ with char $k \notin \{2,3\}$. (Specifically, the number of irreducible $k$-linear representations of $S_4$.) Now I no little of ...
0
votes
0answers
17 views

The fixed field of the Frobenius automorphism

Consider the algebraic closure extension $\overline{\mathbb F_q}/\mathbb F_q$, where $q=p^m$. I wonder if the fixed field of the Frobenius automorphism is $$\sigma: \overline{\mathbb F_q} \to \...
2
votes
3answers
107 views

What group is $\langle (1,1,1) \rangle$ in $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8$?

I know that $\langle (1,1,1) \rangle$ in $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8$ is isomorphic to $\mathbb{Z}_{40}$, but is there a way of writing what group it is (not what it's ...
0
votes
1answer
65 views

My attempt at finding the Galois group of $E:\Bbb Q$, where $E$ is the splitting field of $x^3-5$

I'm trying to understand Galois theory and any help on this question I'm working on would be very much appreciated. Let $E$ be the splitting field of $x^3-5$ over $\Bbb Q$. Compute $\mathrm{Gal}(E:\...
-1
votes
1answer
31 views

Why ignore zero in multiplicative group [duplicate]

As, set of real numbers make group under multiplication but why we ignore zero in multiplicative groups.We can use zero as a counter example to disproof that its inverse doesn't lie in the set??
0
votes
0answers
16 views

Rotman's “Advanced Modern Algebra”, exercise 3.97: a degree function of a Euclidean domain [duplicate]

Rotman defines a degree function of an integral domain $R$ as a function $\delta\colon R\setminus\{0\}\to\mathbb{N}$ so that $(1)$ for any $a,b \in R\setminus\{0\}$ we have $\delta(a) \leq \delta(ab)$...
1
vote
0answers
22 views

The relation between group extension and factor set

I'm new to homological algebra. I try to state what I know, and ask the question at the end. I know that given a group extension $0\to K\to G\to Q\to 1$, it may have many liftings. Each lifting ...
2
votes
1answer
62 views

Using Burnside's lemma for a triangle

Find the number of distinguishable ways the edges of an equilateral triangle can be painted if four different colors of paint are available, assuming only one color is used on each edge, and the same ...
1
vote
1answer
52 views

Extension of basis over PID [closed]

Let $k[x] = R$ be ring and $L$ be free $k[x]$-module; let $v \in L$ be vector in $L$. Then how one can extend it to an $R$-basis for $L$?
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votes
0answers
13 views

sympfying and defining the symmetric difference in sets a ⊕ a [closed]

Define A−B = A∩ B' and A⊕B = (A−B) ∪ (B−A). Simplify each of the following: (i) A⊕ ∅, (ii) A⊕U, (iii) A⊕A, (iv) A⊕A'.
-4
votes
1answer
52 views

The group $S_n$ with a subgroup order $\varphi(n)n$ [closed]

Proof that the group $S_n$ has a subgroup which order is $\varphi(n)n$ and then show that the group $S_{625}$ has subgroup which order is $5^6$.
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votes
1answer
37 views

Is there a procedure to calculate the multiplicative inverse in a quotient by a maximal ideal?

An elementary result in ring theory is that if $R$ is a commutative ring with unity and $M$ is a maximal ideal of $R$, then $R/M$ is a field. There are many proofs of this, as you can see here. But ...
1
vote
4answers
102 views

Uniqueness proof : $a = a'$ so $a$ is unique. Is the proof absolutely rigorous?

My question deals with uniqueness proofs. For example the proof of the uniqueness of the empty set, or the proof of the uniqueness of the identity element in a group. These proofs are convincing of ...
0
votes
1answer
37 views

Polynomials on $\Bbb Z/p^n\Bbb Z$

Please help me with this question: Let $n\in \Bbb N$, let $p$ be a prime number and let $\Bbb Z/ p^n\Bbb Z$ denote the ring of integers modulo $p^n$ under addition and multiplication modulo $p^n$. ...
1
vote
1answer
44 views

Largest ideal inside an ideal of a ring.

Consider a ring $R$ and $I$ be its left ideal. If $I$ is regular, then $(I:R) = \{r\in R\ |\ rR \subset I\}$ is the largest ideal of $R$ that is contained in $I$. I have tried to prove this by ...
0
votes
1answer
23 views

Computing minimal polynomial in finite field F8

In a finite field $F_q$, I've read that one can get the minimum polynomial $f(z)$ of an element $\beta \in F_q$ using this formula: $$f(z) = (z-\beta)(z-\beta^2)(z-\beta^4)(z-\beta^8)...$$ I'm ...
1
vote
0answers
46 views

Why $(1-\omega)^{\varphi(m)} = p\mathbb{Z}[\omega]?$

Let $\omega = e^{\frac{2\pi i}{m}}$ for $m = p^r$ a prime power. Why do we have $$(1-\omega)^{\varphi(m)} = p\mathbb{Z}[\omega]?$$ (Here $(1-\omega)$ means the principal ideal). I know that $$\prod_{...
0
votes
1answer
36 views

If $f^{-1}(j)\leq f(j)$ for all $j$ then $f=f^{-1}$

Let $f$ denote the bijective map from $I=\{1,2,3,4,5\}$ to $I$. If $f^{-1}(j)\leq f(j)$ for all $j$, then $f=f^{-1}$. I have solved the problem using induction. I was wondering if there are some ...
3
votes
2answers
47 views

Homomorphisms from a cyclic group

I'm just starting in abstract algebra, and an exercise in a book sparked the following question. Let's say we want to define a homomorphism $\varphi$ from a cyclic group $G$ (say, $\mathbb{Z}/d\...
3
votes
2answers
47 views

Product of $\prod_{i=0}^{\infty}(1+q^{2^{i}})$ [duplicate]

I know that answer to $\prod_{i=0}^{\infty}(1+q^{2^{i}})$ = $(1+q)(1+q^{2})(1+q^{4})...$ = $\frac{1}{1-q}$. And i need to prove this equality using generating functions. Any hints?
0
votes
1answer
27 views

A question regarding finding the minimal polynomial associated with a field extension . [duplicate]

Say we have the field extension $\Bbb Q(w,\sqrt[3]{5})$ over $\Bbb Q$, where w is the primitive cubed root of unity. I know that the minimum polynomial of $\sqrt[3]{5}$ is $x^3-5$. I want to figure ...
1
vote
1answer
47 views

Generators for $\mathbb{Z_n}$

I would like to show that $K$ is a generator for $\mathbb{Z}_n$ $\iff$ $\gcd(K,n)=1$ and $1 \leq K <n$. My Attempt: Assume $\gcd(K,n)=1$ and $1 \leq K <n$. That means $K \in \mathbb{Z}_n$ and ...
-2
votes
0answers
28 views

There always exists free module [closed]

Prove that if $R$ is an arbitrary ring and $X$ is any set, then there exists a free module on $X$. Definition of free module I use: Given a (possibly empty) set $X$, a $R$-module $M$ and a function $...
1
vote
0answers
19 views

relation between semisimple lie algebra completely reducibility and semisimple ring

Let $g$ be a semi-simple finite dimensional lie algebra over complex number $C$. Then every non irreducible representation of $g$ is completely reducible. $\textbf{Q1:}$ Is category f.g. left $U(g)-...
0
votes
2answers
36 views

Proof $F$ is a free module on $X$

Let $\{X_i:i\in I\}$ be a collection of mutually disjoint sets and for each $i \in I$ let $F_i$ be a free module on $X_i$ with $l_i: X_i\rightarrow F_i$. Let $X=\bigcup_{i\in I}X_i$ and $F=\sum_{i\in ...
1
vote
2answers
52 views
+50

Can Rings be viewed as “function rings over their spectrum”?

In the nlab article about localization, a side note says When interpreting a ring under Isbell duality as the ring of functions on some space $X$ (its spectrum), […] Unfortunately, the article ...
-1
votes
1answer
42 views

Non-zero ideals in ${\mathbb{Q}}_p$ are $p^n{\mathbb{Q}}_p$, $n\in\mathbb N_0$

How do I show that every non-zero ideal in ${\mathbb{Q}}_p$ is of the form $p^n{\mathbb{Q}}_p$ for some $n \in \mathbb{N}_0$, and investigate if ${\mathbb{Q}}_p$ is a principal ideal domain? If it ...
0
votes
1answer
26 views

Free module definition

Let $X$ be non empty set and let $F$ be a left $R$ module. Left $R$ module with function $f:X\rightarrow A$ is called free on $X$ if exists unique homomorphism of $R$ modules $g: F\rightarrow A$ such ...
0
votes
1answer
18 views

Let $\phi: G\rightarrow G'$ be a homomorphism. Is $Ker(\phi) =\phi^{-1}(e')$?

Let $\phi: G\rightarrow G'$ be a homomorphism. Is $Ker(\phi) =\phi^{-1}(e')$? I say yes it is. But someone told me once that the inverse image is not necessarily in $G$ so I feel like there may be ...
1
vote
1answer
36 views

Prove the following for a general binary operation

I need some help with the following proof, please. First, a definition below: A binary operation $p$ on a set $X$ is a function of two variables, whose values lie in $X$: it assigns to each ordered ...
0
votes
0answers
23 views

Quotient of module versus direct sum

It's maybe stupid question but I want to be sure about what I've learned. Is the quotient $R$-module $R/(p^n) \simeq R/(p) \oplus...\oplus R/(p)$ in general, if not when? I think over $\mathbb{Z}$-...
0
votes
0answers
16 views

A problem related to the definition of group extension

Given a group extension, wikipedia said that the group $N$ is isomorphic to a normal subgroup in $G$. However, how can I deduce it? Can't see the reason directly. Or am I misunderstand it?
0
votes
0answers
17 views

What does free left $\Bbb ZG$ module with basis $G^n$ mean?

I'd just stuck in a sentence Rotman said in his Advanced Algebra book mention a word "free left $\Bbb ZQ$ module with basis $Q^n$". However, as far as I know, a free $R$-module on a set $X$ is $\...
5
votes
3answers
109 views

Finding degree of a finite field extension

Let $x=\sqrt{2}+\sqrt{3}+\ldots+\sqrt{n}, n\geq 2$. I want to show that $[\mathbb{Q}(x):\mathbb{Q}]=2^{\phi(n)}$, where $\phi$ is Euler's totient function. I know that if $p_1,\ldots,p_n$ are ...
0
votes
1answer
44 views

Prove that regular values are dense (Brown-Sard Theorem), is true by showing it is true for an open subset.

I don't believe this is necessarily related to cohomology. This could be something about manifolds in general. My book is From Calculus to Cohomology by Ib Madsen and Jørgen Tornehave. I. The claim ...
0
votes
1answer
11 views

Notation for this quaternary linear code

I've been reading about self dual codes, and the literature says that there is a quaternary self dual code $$i_2 \otimes \mathbb{F}_4$$where $i_2 = {\{00,11}\}$ is a binary self dual code and $\mathbb{...
2
votes
0answers
54 views

Two commutator relations.

$\newcommand{\ad}{\operatorname{ad}}$Let $R$ be an associative ring. Set $[x, y] = xy - yx$ and $\ad_x(-) = [x, -]: R \to R$. Is there a formula for $(ab)^n$ in general? I found one formula for the ...
-1
votes
1answer
29 views

F has a nonempty basis implies F is the direct sum of a family of infinite cyclic subgroups [closed]

Let $F$ be an Abelian group. How to prove the statement in the title?
2
votes
3answers
58 views

What is the difference between using maximal ideals to define Zariski topology versus using prime ideals?

I just started looking at the notes https://www.jmilne.org/math/CourseNotes/iAG200.pdf. And in the Appendix where they review some algebraic geometry they define sets of the form $$ Z(\mathfrak{a}) = ...
3
votes
1answer
20 views

Showing that an automorphism of $S_4$ fixing each Sylow 3-subgroup must be the identity.

I am working on a Hungerford exercise and trying to show that $S_4$ is isomorphic to its automorphism group. I know that there are four Sylow 3-groups in $S_4$ and the four Sylow groups exhaust the 3-...