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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.

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0answers
7 views

ow many different ways can one arrange eight place settings at a round dinner table, using up to three different colors of dinner plate?

Using the Burnside Theorem, how do i solve the following problem? How many different ways can one arrange eight place settings at a round dinner table, using up to three different colors of dinner ...
1
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1answer
16 views

Tensor products of rings as modules over themselves

Let $R$ be a non-commutative ring, and, consider the tensor product $$R \otimes_R R $$ where we consider the 'first $R$' as a right module over $R$ and the 'second $R$' as a left module over itself. ...
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votes
2answers
29 views

Multiplication of two cyclic groups [on hold]

Let A and B be two cyclic groups of order m and n respectively. Show that A×B need not be a cyclic group.
1
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1answer
27 views

Checking of normal subgroup

Show that if $H$ is the only subgroup of order $n$ in a group $G$ then $H$ is a normal subgroup of $G$.
1
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1answer
29 views

Understanding a classical theorem on commutator subgroup

I don't get the point when I see the theorem in the below figure. What does $G'\leq\ker\varphi$ want to tell (I know the mathematical meaning word by word, but don't know the intuitive meaning)? What ...
2
votes
1answer
23 views

Stuck in a step of the deduction of cocycle identity

In the proof of the cocycle identity, how does the red arrow valid? I'd thought about it for a long while, but can't see the reason. PS: The author denote the operation in $G$ as $+$ for convenience (...
1
vote
1answer
28 views

Understanding the proof of a criterion of equivalent group extensions

I stuck in a step of the proof of the equivalence of two group extensions for 30 mins. In the place of red arrow, how does the previous line deduce the next? PS: The author denote the operation in $G,...
0
votes
1answer
19 views

Existence of submodules [duplicate]

Show that if $R$ has an identity and $A$ is a $R$-module, then there are submodules $B$ and $C$ of $A$ such that $B$ is unitary, $RC = 0$ and $A = B ⊕ C.$ Give my hint, I don't know how to start.
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1answer
18 views

Checking Group isomorphism

Let G be a group and A,B be normal subgroups of G such that A is isomorphic to B.Show by an example that G/A need not be isomorphic to G/B.
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1answer
34 views

Is $B \otimes_A M$ a free $B$ module if $M$ is a free $A$ module?

Let $A\subset B$ be commutative rings with identity. Let $M$ be a free $A$ module. Then $B \otimes_A M$ is a $B$ module. It is also a free $A$ module . But is it a free $B$ module?
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0answers
17 views

Generator matrix of Reed Solomon code [on hold]

Given the (10, 3, 8) Reed-Solomon code over $F_{11}$. My question is simple: How can I construct the generator matrix?
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0answers
14 views

Generator matrix for classical binary Goppa code [on hold]

I've given $F_8$ which is equal to $F_2 (\alpha)$, where $\alpha$ is a root of $x^3 + x+1$. How can the generator matrix for a classical binary Goppa code $T(L, G)$ be constructed with Goppa ...
0
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0answers
30 views

An equation that seems like an eigenvalue equation but it's probably not

Given a unitary $n\times n$ matrix $U$, is it possible to find a normalized vector $v$ and a scalar (complex) function $g(U)$ such that $$ Uv = g(U)v $$ holds? As far as I understand, the equation ...
0
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3answers
26 views

How many elements of have square roots in a field of 13 elements? [duplicate]

Initially, I thought since this field was isomorphic to $({0,...,12})$ , the elements $(0,4,9)$ would have square roots. However, when I checked the solutions, the answer was different. Thank you in ...
0
votes
0answers
17 views

Any $f': A \to Hom_{\mathbb{Z}}(R,M)$ can be lifted to $F': B \to Hom_{\mathbb{Z}}(R,M)$ $R$ module homomorphism with $f'=F' \circ \psi$

let $\psi: A \to B$ be an injective $R$ module homomorphism, and it is given that any $f: A \to M$ $\mathbb{Z}$ module homomorphism can be lifted to a $\mathbb{Z}$ module homomorphism $F: B \to M$ s.t ...
0
votes
2answers
38 views

A question about the rules of modular arithmetic .

If we have a modular equation say $5^{x+1}\mod2^{n+1}$, where $5^x \equiv1\mod2^n$, we also know that $5\equiv1\mod2$. I know from playing around with an online mod calculator that $5^{x+1}\equiv1\...
0
votes
0answers
32 views

diagonalize general quadratic form

I want to diagonalize an arbitrary quadratic form $$\sum_{i\leq j \leq n} a_{i,j}x_ix_j$$ over an algebraically closed field of characteristic not equal to $2$. If $x_n^2$ appears in the form with ...
0
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1answer
26 views

How do I use Cayley's Theorem to find a subgroup that is Isomorphic to a Group? [duplicate]

Let $G = \mathbb Z/2\mathbb Z × \mathbb Z/2\mathbb Z$. Use Cayley’s theorem to determine explicitly a subgroup of $S_4$ that is isomorphic to $G$.
0
votes
1answer
38 views

Show that $\psi \circ \phi$ is an isomorphism.

Let $\phi : G_1 \to G_2$ and $\psi : G_2 \to G_3$ be isomorphisms. Show that $\phi ^{-1}$ and $\psi \circ \phi$ are both isomorphisms. Using these results, show that the isomorphism of groups ...
1
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0answers
34 views

Classical $\sum_{1\leq i\leq n} x_i^2$ are group forms for $n=1,2,4,8$.

Let $F$ be a field. Consider quadratic forms $f=\sum_{1\leq i\leq n}x_i^2$ with $n=1,2,4,8$. $f$ is a group form if $\{d\in F^\star|\exists x\in F^n, f(x)=d\}$ is a group. For $F=Q$, it follows from ...
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0answers
22 views

Characterizing a module of Kahler differentials

Consider the $\mathbb C$-algebra $R=\mathbb C[x,y,z]/(z(y^2-x^3)-1)$. How to prove that the module of Kahler differentials $\Omega_{R/\mathbb C}$ of $R$ over $\mathbb C$ is a free $R$-module of rank 2?...
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1answer
39 views

How to check $x^2+y^2+z^2=7 w^2$ admits no no-trivial integral solution?

This is a statement made in Lam's Introduction to Quadratic Forms Chpt 1, Sec 2. "7 is known not in $D(f)$ in elementary number theory" where $D(f)=\{(x,y,z)\in Q^3, x^2+y^2+z^2=7\}$ and $Q$ is ...
3
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0answers
31 views

Prove that every prime ideal that isn't maximal is a minimal prime ideal

Suppose that the additive group of the ring $R$ is a finitely generated abelian group. If $P$ is a maximal ideal of $R$, show that $R/P$ is a finite field. Show that every prime ideal of $R$ that is ...
0
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1answer
30 views

Find the amount of subgroups of order $3$ and $21$ in non-cyclic abelian group of order $63$

Find the amount of subgroups of order $3$ and $21$ in non-cyclic abelian group of order $63$. In first case I found the amount of elements that have order $3$ - there are $8$ of them, in second case ...
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4answers
55 views

Prove $+$ and $\times$ are well-defined on quotient rings

I am asked to prove that the $+$ and $\times$ operations are well-defined on quotient rings. Having little experience with similar questions I looked up the proofs online and found that most of them ...
2
votes
0answers
10 views

Methods of verifying the Smith Normal Form of a matrix

I have an algebra exam in a few weeks and, if the past papers are anything to go by, it seems likely that there will be a question on finding the Smith normal form of a 4x4 matrix with entries in $\...
3
votes
1answer
51 views

For any finite abelian group $G$, there is an integer $m$ with $G$ isomorphic to a subgroup of $U(\mathbb{Z}_{m})$.

I want to prove if the following assertion from Rotmans Advanced Algebra page 205 is true: For any finite abelian group $G$, there is some integer $m$ with $G$ isomorphic to a subgroup of $U(\mathbb{...
1
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2answers
34 views

Let J be an ideal. Find a function in I(V(J)) such that the function f is not in J

Let $J$ be the ideal $\langle x^2+y^2-1,y-1\rangle$. Find $f \in \textbf{I}(\textbf{V}(J))$ such that $f \not \in J$. I'm confused on a number of aspects here. Firstly, how do I find $\mathbf{V}(J)$ ...
1
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1answer
17 views

Why a particular module over a DVR is cyclic if it's quotient and kernel are cyclic?

I'm looking at this lemma below. Here $A$ is a DVR with uniformizer $\pi$. I'm not sure how to justify the underlined step. Why does $M_n$ have to be cyclic if $M_1$ is cyclic?
0
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2answers
25 views

Real projective space and $n$-sphere

I know that $$\mathbb{R}^n\mathbb{P}\cong \mathbb{S}^n/{\pm 1}$$ Is there another equivalence relation apart from treating antipodal points as the same which we can quotient out from $\mathbb{S}^n$ to ...
1
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2answers
38 views

Is this modest change enough to make an idempotent element an identity element?

In The Number System by Thurston the author introduces an algebraic structure he calls a hemigroup. The laws of a hemigroup are: (i) $\left(x\odot y\right)\odot z=x\odot \left(y\odot z\right),$ (...
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2answers
26 views

How to understand $\mathbb{Z}^n$-graded ring?

I am reading Ringel's note and I encountered a question I have never met. The question is what's the meaning of a $\mathbb{Z}^n$-graded ring? This is from the following: "Note that the rings $U_q (n_ ...
2
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2answers
108 views

Determining the ideals of a quotient ring

Given an ideal $I = \langle x^3 - x\rangle \subseteq \Bbb{R}[x]$, determine the ideals in the quotient ring $\Bbb{R}[x]/I$. I understand that the quotient ring is of the form $k[x_1...x_n]/I$ where ...
2
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0answers
37 views

Is there a formal term for the algebraic structure equivalent to Thurston's hemigroup without identity? [duplicate]

This question is not a duplicated of https://math.stackexchange.com/a/3183330/342834 . It is a follow-on, asking about the distinction between the subject of that post, and the subject of this post. ...
0
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1answer
23 views

What does “Realizes the operator” really mean?

Let $K,~Q$ be two groups, $K$ is abelian and $K$ is also a $\Bbb Z[Q]$-module. Then a group extension of $K$ by $Q$: $0\to K\to G\to Q\to 1$ realizes the operator if the scalar multiplication $Ck=\ell(...
0
votes
1answer
50 views

Enigma having to do with bricks

A professor told us this one but I can't remember much of it, I hope you can help me find it. This is what I remember, you have bricks, each brick has only (or at least) a side which length is an ...
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0answers
42 views

Group theory and Schur’s lemma

I am a little bit confused by consequence of Schur’s lemma: “irreducible representation of any abelian group must be of dimension one”. The proof is quite simple but i have some doubts: for exemple if ...
1
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1answer
17 views

Showing W=ku+kv has a unique module structure over kG

Let $G=<x> \times <y>$ where $|x|=|y|=p$ and so $|G|=p^2$. Let $k$ be a field of characteristic $p$. Let $W$ be the $k$-span of $v$ and $u$. We wish to show the module structure given by $...
0
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1answer
21 views

Proving that if coprime $\alpha_{i}\in R$ divide b, then $\alpha_{1}…\alpha_{n}$ divide b.

Let $R$ be a principal ideal domain and let $\alpha_{1},...,\alpha_{n}\in R$ be such that $(\gcd(\alpha_{i},\alpha_{j}))=(1)$. Let $b\in R$ such that each $\alpha_{i}|b$. I want to show that $\...
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3answers
46 views

Why we never use the product between vectors like between elements of direct groups product?

We can say, that any field $\mathbb{K}$ -- $1$-dim vector space on itself: $\mathbb{K}_{\mathbb{K}}$. So any vector of one another finite-dimensional vector space $V_{\mathbb{K}}$, after choosing the ...
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1answer
50 views

Prove that if $G$ is a $p$-group, $p$ is prime, then $G'=[G,G] \subseteq Z(G)$

I have proved if $G$ is a $p$-group, $G$ is nilpotent, and the lower central series and the upper central series have a finite length and both cover from $G$ to/downto $\lbrace 1 \rbrace$. But I do ...
2
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0answers
34 views

Proof of : “ Union of all R-equivalence classes on A is included in A” based on “ R is a subset of A² ”. ( Q° on Ayres, Problems Modern Algebra)

I'm studying Ayres Theory and Problems of Modern Algebra. In Chapter 2, Solved Problem 6, the question is asked to prove that " An equivalence relation R on a set S effects a partition of S." ...
3
votes
1answer
71 views

$A\otimes \mathbf{Q}\cong B\otimes \mathbf{Q}$ implies $A$ and $B$ are $\mathcal{C}$-isomorphic

I am trying to solve exercise 211 on Davis-Kirk: Let $\mathcal{C}$ be the class of torsion abelian groups. Show that for any abelian groups $A,~B$, $A\otimes \mathbf{Q}\cong B\otimes \mathbf{Q}$ ...
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votes
1answer
40 views

Let $k$ be the largest order of elements of finite abelian group A. Prove, that the order of any element in $A$ divides $k$. [on hold]

Let $k$ be the largest order of elements of finite abelian group A. Prove, that the order of any element in $A$ divides $k$. Finite abelian group A is isomorphic to $\mathbb{Z}_{p_{1}^{t_{1}}} \...
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0answers
27 views

Difference between blowup of $V_1\cup V_2$ and union of blowup of $V_1$ and blowup of $V_2$

Let $V_1,V_2\subset \mathbb C^N$ be sub varieties and let $C=\{f_1=\dots=f_d=0\}$ be a subvariety of $\mathbb C^N$. Consider the blowup of $\mathbb C^N$ along $C$ and denote it by $B_C\mathbb C^N$. ...
0
votes
0answers
22 views

asymptotically left uniformly continuous and left uniformly continuous functions on semigroup S

Given a non-empty set $S$, we denote by $L^{∞}(S)$ the Banach space of bounded real-valued functions on $S$ with the supremum norm. Let $S$ be a semigroup. Then a subspace X of $L^{∞}(S)$ is left ...
0
votes
1answer
34 views

How can I prove that $H_1/N_1 \ncong H_2/N_2$?

So I have a non-surjective homomorphism $\phi: H_1 \to H_2$, and $N_1 \unlhd H_1$ and $\phi(N_1) = N_2$. How do I prove that $H_1/N_1 \ncong H_2/N_2$? All I have right now is an example that works, ...
0
votes
0answers
44 views

Is $H/N$ a quotient group even if $N$ is just a subgroup of $H$ and not necessarily normal?

I am very confused about the notation of $H/N$ and whether it always implies a quotient group. The confusion stems from the following statement of the fourth isomorphism theorem: Let $G$ be a group ...
0
votes
1answer
22 views

Showing right-exactness of a sequence between modules

I am trying to show the right-exactness of a sequence of a sequence of modules, but I am stuck at the last step. Let $R$ be a ring, and $L, M,N$ be $R$-modules (In the exercise I am trying to solve ...
4
votes
2answers
197 views

Does every subgroup of an abelian group have to be abelian?

My original problem is to show that E/L is an abelian extension over L and L/F is an abelian extension over F, given that E/F is an abelian extension over F and that L is a normal extension of F such ...