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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.

0
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1answer
63 views

What is $\deg(f)$ when $\int_M \omega = 0$, if that's possible?

In this question, Jrrow assumes $\int_N\omega_0\neq 0$ Why is $\deg(f)$ well-defined? What if $\int_N\omega_0\neq 0$? My book does not seem to address this explicitly. If $\int_N\omega_0 = 0$, then $...
3
votes
4answers
55 views

Elements of order 2 in $D_{2n}$

Im new at this abstract algebra stuff and im not comfortable with the proofs techniques yet, so I have a question related to the elements of order $2$ in $D_{2n}$. Problem: Prove that $\{x\in D_{2n}...
1
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2answers
46 views

Proving the given $\mathbb R^3/H$ $\cong$ $\mathbb R^2$ where $H$ = {$(y,0,0)|y \in \mathbb R$}

So I am given a group $\mathbb R^3$ and a group $H$ = {$(y,0,0)|y \in \mathbb R$}. I have to prove that that $\mathbb R^3/H$ $\cong$ $\mathbb R^2$. I am not sure how to even begin. My difficulty is ...
0
votes
0answers
16 views

Prove that $A_n$, the set of even permutations on a set of length n, is a transitive group of transformations on $S=\{1,2,3,…,n\}$

I already know that $S_n$ is a transitive group of transformations on $S=\{1,2,3,...,n\}$. I am sure that it has to be with the fact that $A_n \subset S_n$ but I do not how to proceed.
1
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0answers
24 views

Semigroup and their monogenic semigroup

A subsemigroup $K$ of $S$ is said to be monogenic if $K = \langle a \rangle$ for some $a \in S$ and the order of $a$ is the size of $K$. If the order of $a$ is finite, then after some time power of $a$...
0
votes
0answers
29 views

What is meaningful product?

Sentence is this. The product of a standard n product and a standard m product is a meaningful product equal to the standard (m+n) product. This sentence is in Thomas W. Hungerford Algebra : Groups ...
1
vote
1answer
58 views

Is $\lim\limits(\prod_{i\leq n} K)\cong\cup_i(\prod_{j\leq i}K)$ true?

Consider a family of $K-$vector spaces $\prod_{i\leq n}K\to\prod_{i\leq n+1}K$ by embedding where embedding is done by identifying the basis $(1,0,\dots,0)\to (1,0,\dots,0,0),...$ and similarly for ...
6
votes
2answers
97 views

Does it suffice to check the normal subgroup property for the generators?

Let $G$ be a group generated by a subset $S$ and $H$ be a subgroup of $G$ generated by a subset $T$. To check whether $H$ is a normal subgroup of $G$ or not, we must check the following statement: $$ ...
5
votes
1answer
90 views

$\operatorname{Frac}(A)/A$ as an $A$-module

I am wondering about a question: We know that $\mathbb{Q}/\mathbb{Z}$ is torsion group and $\mathbb{Q}/\mathbb{Z}=\bigoplus_{p\text{ prime}}\mathbb{Q}/\mathbb{Z}(p)$ where $\mathbb{Q}/\mathbb{Z}(p)=\{...
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0answers
41 views

No simple groups of given order.

I am trying to show the following: Prove that there are no simple groups of the given order: 42. 200. 231. 255. I understand that they need to be broken down into their prime factors. I was ...
0
votes
1answer
42 views

Number of different quadratic functions mod 12.

How many different quadratic functions are there of form: $x \mapsto ax^2+bx+c \pmod{12}$ All I could come up with is an upper bound of $11*12*12$. This is a puzzle from a YouTube video by ...
1
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2answers
43 views

Nilpotent Ideal

I read the definition of Nilpotent Ideal but having hard time grasping it. I need some simple examples to feel what the definition meant.
0
votes
0answers
31 views

Equality of morphism of representations of $S_n$ and $A_n$

Let $S_n$ be the symmetric group and $A_n$ the Alternating group. Let $(V,\rho)$ be a simple representation of $S_n$. Let $(W,\pi)$ be a simple representation of $A_n$. Suppose $V=W$. I saw in some ...
0
votes
1answer
12 views

Bijection between the coset of the Young-subgroup and (p,q)-shuffles

Let $n=p+q$ and $I$ be a subset of the natural numbers up to $n$ with $p$ elements. I have to show, that there is a bijection between $S_n / Y_I$ where $Y_I$ is the Young-subgroup and all $(p,q)$-...
-1
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0answers
38 views

Proof to show that a ring is a field

is (3Z, +, •) a field? I have done most of the proof except for the part of zero divisors and non-zero zero divisors. Furthermore, is this an integral domain?
1
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2answers
47 views

Notation issue with 'Show that $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$.'

Show that $\mathbb{Q}[\sqrt{2}]$ is a vector space over $\mathbb{Q}$. I haven't seen this notation $\mathbb{Q}[\sqrt{2}]$ before. Can someone clarify what it means?
3
votes
1answer
46 views

Ideals in polynomial ring

I am facing this problem: Prove that there is a bijection between the monic divisors of $x^n−1$ in $F[x]$ and the ideals of $F[x]/\left<x^n−1\right>$. I tried to find how the ideals in $F[x]/...
4
votes
1answer
76 views

Is there a good way to show that the order of element in $S_7$ are at most $12$?

The only solution would be going through all cycle types of all permutations which is a lot of work. Is there any smarter solution than this one? Thank you in advance!
1
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1answer
25 views

Automatic theorem proving with term rewriting on finitely presented algebraic structures

I am looking for an open source software package that can do automatic theorem proving for finitely presented algebraic structures. You should be able to program in the axioms of the structure and ...
0
votes
0answers
91 views

How does a group look like in its symmetric group?

(This may look similar to past questions of mine, but actually adds something and goes more straight to the point.) Let $G$ be a (possibly infinite) group. Left and right multiplications establish ...
-1
votes
0answers
15 views

Peterson-Gorenstein-Zierler algorithm [closed]

Given: $\alpha$ as a primitive element of $F_{2^4}$ satisfying $\alpha^4 + \alpha + 1 = 0$ and the [15,7,5] binary BCH code given by the generator polynomial $g(x) = x^8 +x^7+x^6+x^4+1 = \prod_{i \in ...
3
votes
0answers
35 views

Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$? [duplicate]

Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$? It is rater obvious, that if such group exists then it is generated by any ...
3
votes
2answers
97 views

Show that $\pi_1(M)=\langle a,b|b^{2}=1 \rangle\cong \Bbb Z*\Bbb Z_2$ is non-abelian.

How to show that $\pi_1(M)=\langle a,b|b^{2}=1 \rangle\cong \Bbb Z*\Bbb Z_2$ is non-abelian by taking any two elements of this group that don't commute? Thanks in advance!
0
votes
1answer
17 views

BCH code with received vector

I've got $\alpha$ as a primitive element of $F_{2^4}$ satisfying $\alpha^4 + \alpha + 1 = 0$ and the [15,7,5] binary BCH code given by the generator polynomial $g(x) = x^8 +x^7+x^6+x^4+1 = \prod_{i \...
0
votes
1answer
24 views

Induced map between associated graded algebra?

I have just learned how one can take an filtered algebra and get an associated graded algebra, (here's the wikipedia article for reference). I have also seen in various places, (such as this question),...
-3
votes
0answers
25 views

A mapping problem.

To show onto. Choosen any element $c$ from $U(q')$, there exists a pre image in $U(q)$ such that the given relation holds. But I can't frame it out. Can anyone suggest how to proceed. Thanks in ...
1
vote
1answer
30 views

Noncyclic subgroups of multiplicative group of integers mod n

I want to find an $n$ such that $\mathbb{Z}/n\mathbb{Z}^{\times}$ has a noncyclic subgroup, and I'm struggling to think of an example of such a subgroup. How can I construct such a subgroup without ...
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0answers
16 views

Abstract algebra in pmi [closed]

Use the principle of mathematical induction to show that if S is a set with n elements, then it has n 2 subsets, ∀ n .
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2answers
30 views

Example of a field of characteristic 0 and a finite multiplicative subgroup of order 10?

Let $G$ be a finite subgroup of the multiplicative group of some field $F$. I want to find an example of a field $F$ of characteristic 0 and a $G$ of order 10. I can think of a number of fields of ...
0
votes
2answers
25 views

which elements of $\mathbb{Z}_n$ give the entire ring?

Which a ∈ $\mathbb{Z}_n$ satisfy 〈a〉$_\mathbb{Zn}$=$\mathbb{Z}_n$ From other theorems I'm thinking that $a_n$ must be a unit, but I don't know how to prove it... or if that is even correct.
2
votes
1answer
35 views

Let $G$ be a nilpotent group prove that for each $x \in Z_2(G)$ the map $\theta_x: G \rightarrow Z(G)$ defined by $\theta_x(g)=[g,x]$ is a hom

Let $G$ be a nilpotent group of class c. Prove that for each $x \in Z_2(G)$ the map $\theta_x: G \rightarrow Z(G)$ defined by $\theta_x(g)=[g,x]$ is a homomorphism with kernel $C_G(x)$. A hint is ...
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0answers
46 views
0
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0answers
51 views

Does $X/K\cong Y/K$ imply $X\cong Y$ for modules?

Let $X$ and $Y$ be left modules over a ring $R$, and let $K$ be a submodule of both $X$ and $Y$ such that $X/K\cong Y/K$. Does this imply that $X\cong Y$? In the particular case that $X$ and $Y$ are ...
0
votes
0answers
34 views

Erroneous Definition of Stem Field

Here's the definition of stem field provided by the book I am using (https://www.jmilne.org/math/CourseNotes/FT.pdf): Let $f$ be a monic irreducible polynomial in $F[X]$. A pair $(E,\alpha)$ ...
0
votes
2answers
27 views

Product of a class function with a conjugate of an irreducible character

Let $\chi$ be an irreducible complex character of a finite group $G$ and define $f:G \to \mathbb{C}$ by $f(g)=|\{h \in G:h^2=g\}|$. From a question which I am trying to solve it appears that $f(g)\...
2
votes
1answer
45 views

Least period of the Fibonacci sequence in a field

Actually, I'm solving some exercises from the book "Finite Field" by Rudolf Lidl et al. There is an exercise for which the idea is missing to solve it: Let $r$ be the least period of the Fibonacci ...
2
votes
1answer
40 views

Prove that H is a group under multiplication

I want to show that H is a group under multiplication where $H=\{a^x|x \in \mathbb{R}\}$ To show it's a group then I must show associativity, identity, inverse, and closure. For Associativiy: Let $...
1
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0answers
44 views

For an $A$-module $T$ and $T_{0} \in \operatorname{Add}(T)$, when is $\operatorname{Ann}(T) = \operatorname{Ann}(T_{0})$?

Let $A$ be any unitary ring, let $T$ be an $A$-module and $T_{0} \in \operatorname{Add}(T)$. Clearly, $\operatorname{Ann}(T) \subseteq \operatorname{Ann}(T_{0})$. As the title says, I'm wondering ...
0
votes
0answers
16 views

$|G| = n$ and $k | n$, then $G$ has a subgroup of order $k$ [duplicate]

After thinking it a while and since I wasn't able to think how to prove it, I started thinking that this might be false. At first I had to show if $|G| = n$ and $k | n$, then $G$ has an element of ...
0
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2answers
38 views

Showing $x^m$ is an automorphism

Let $m,n \in \mathbb N$ and they are relatively prime and $G$ is a group with $|G|=n$. Show that $T:G \to G$ is an automorphism of $G$ where $T(x)=x^m$ First I showed it's a homomorphism. I have ...
0
votes
0answers
28 views

For an equivalent functor $\Sigma:T \to T$, if $T(v \circ u)=0$ then $v \circ u=0$.

Im beginnig selfstudy of triangulated categories, and Im working with an additive category $T$ and an additive covariant and equivalent functor $\Sigma:T \to T$, as equivalent the notes say that ...
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0answers
28 views

Need help understanding an equation: composition, addition and inverse

I have found an interesting paper on a digital image registration algorithm. There are many equations in the paper that I only understand partially, but there is a particular one I would like to ...
3
votes
0answers
23 views

Is there a formula for $[F_n : V_{\{x^3\}}(F_n)]$?

Suppose $F_n$ is a free group of rank $n$. It is a rather well known fact, that $b_3(n) = [F_n : V_{\{x^3\}}(F_n)]$ is finite for all $n \in \mathbb{N}$. Is there a some sort of formula for $b_3(n)$? ...
0
votes
0answers
28 views

Why is $GF(p^n)$ unique? [duplicate]

I'm having trouble understanding why exactly $GF(p^n)$ is unique up to isomorphism. I know that the proof begins by claiming that $GF(p^n)$ is the splitting field of the polynomial $x^{p^n}-x\in \...
0
votes
2answers
26 views

Units in tensor products of commutative algebras

Let $A$ be a commutative algebra of finite dimension over a field $F$. Then $A\otimes A$ is also a commutative algebra. Clearly if $u_1,u_2$ are units in $A$, then $u_1\otimes u_2$ is a unit in $A\...
0
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0answers
35 views

Ring Homomorphism unit to unit mapping

Fraleigh defines a ring homomorphism $\phi:R \to R'$ as a map $\phi$ such that $$\phi(x+y) = \phi(x) + \phi(y)$$ and $$\phi(xy) = \phi(x) \phi(y).$$ As such, it must follow directly from the ...
1
vote
2answers
27 views

Quotient of non-commuting indeterminates isomorphic to commuting indeterminates

Let $K$ be a field. I am trying to prove that the quotient of the space $K\langle x,y\rangle$ of two non-commuting indeterminates $x,y$ with $I=(xy-yx)$, that is, the two-sided ideal generated by $xy-...
3
votes
1answer
55 views

Finding the order of $\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle.$

Im new at abstract algebra stuff and im wondering whats the technique to prove this kind of stuff. Question: Let $G=\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle$, prove that $|G|=16 $ and find all ...
0
votes
0answers
41 views

Irreducible complex characters of a finite group

Let $G$ be a finite group. If $\chi$ is a complex character of $G$, we define $\overline{\chi}:G \to \mathbb{C}$ by $\overline{\chi}(g)=\overline{\chi(g)}$ for all $g \in G$. We write $\nu(\chi):= ...
1
vote
0answers
29 views

A sequence of polynomial is irreducible or not under applied condition

Which of the following statement is/are true for the following equation, where $n$ is a positive integer? $$f_{n}(x)=x^{n-1}+x^{n-2}+\cdots+x+1$$ $f_{n}(x)$ is irreducible over $\mathbb{Q(x)}$ ...