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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

Group’s 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
9 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 $...
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1answer
14 views

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

Let $R$ be a principle 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
36 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
43 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 ...
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0answers
19 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
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1answer
40 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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1answer
33 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
15 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 $B_C\mathbb C^N$. ...
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0answers
16 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 ...
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1answer
29 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, ...
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0answers
41 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 ...
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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 ...
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2answers
187 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 ...
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1answer
33 views

proving $x$ is the generator of a cyclic group

Show that $x$ is a generator of $(\mathbb{Z}_3[x]/\langle x^3+2x+1\rangle)^*$. I don't understand part of the solution. $x^3+2x+1$ is irreducible in $\mathbb{Z}_3$. Let $a$ be a zero of $x^3+2x+1$ in ...
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2answers
41 views

Find the smallest $n \in \mathbb{N}$ such that the group is isomorphic to the direct product of $n$ cyclic groups

Find the smallest $n \in \mathbb{N}$ such that the group $\mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45}$ is isomorphic to the direct product of $n$ cyclic groups. I'm not sure but if I ...
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0answers
16 views

The multiplicative group of units in $\mathbb{Z}_{p}$ is isomorphic to $\mathbb{Z}_{p} \times C_{n}$ with $n=\max\{p-1,2\}$.

Problem. (a) Write down explicitly the rules for addition and multiplication in $\mathbb{Z}_{p}$. (b) Show that $(\mathbb{Z}_{p^{i}},\varphi_{ij})$ where $\varphi:\mathbb{Z}_{p^{j}} \to \mathbb{...
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0answers
55 views

How to proceed with following problem of algebra [on hold]

How to proceed with this problem. I tried to form equation but it cant be solve with that. $$\begin{align}\dfrac{a^2}{2^2 - 1^2} + \dfrac{b^2}{4^2 - 1^2} + \dfrac{c^2}{6^2 - 1^2} + \dfrac{d^2}{8^2 - ...
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0answers
12 views

Generator matrix of Reed-Solomon code [on hold]

Given the [10, 3, 8] Reed-Solomon code over $F_{11}$ with evaluation points $\alpha_i = i$ for $i = 1, 2, ..., 10$. How can I construct, with these information, the generator matrix?
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0answers
15 views

Inverse limit of projective profinite groups is projective

I'm trying to prove the following (H.W question): Let $ G $ be inverse limit of projective profinite groups. Prove that $G$ is projective group. Projective group means "cohomological dimension ...
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0answers
14 views

Generator matrix for binary Goppa code

Let $F_8 = F_2 (\alpha)$, where $\alpha$ is a root of $x^3+x+1$. I'd like to construct the generator matrix for a binary Goppa code $T(L, G)$ with Goppa polynomial $G(x) = x+1 \in F_8[x]$ and L the ...
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votes
1answer
21 views

Surjectivity of a bilinear map between groups

Let $G$ a generic group and $\gamma_i(G)$ its lower central series defined as $$ \begin{cases} \gamma_1(G)=G\\ \gamma_{i+1}(G)=\left[\gamma_i(G); G\right] \end{cases} $$ for every $i\in \mathbb N$. ...
2
votes
1answer
35 views

How to show that some elements of $\mathbb{Z}[2\sqrt{2}]$ are irreducible?

I want to show that $2$ and $2\sqrt{2}$ are irreducible in $\mathbb{Z}[2\sqrt{2}]$. Consider the norm $N:\mathbb{Z}[2\sqrt{2}]\to\mathbb{Z}_{\ge0}$ defined by $N(a+b\cdot2\sqrt{2})=a^{2}-8b^{2}$. ...
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votes
0answers
13 views

Effect of puncturing and shortening codes [on hold]

Let $C$ be a linear [n, k, d] code. Then, $\bar{C} = [n-1, \bar{k}, \bar{d}]$ is an i-punctured code of C. (The i-th coordinate is deleted.) And $C' = [n-1, k', d']$ is an i-shortened code of C (...
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1answer
25 views

Help understanding a theorem about group homology $H_0$

I'm self-studying homological algebra. I have problems on understanding a theorem about $H_0$. First, I don't know where the bottom row of the commutative diagram comes from, see the red line in ...
1
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1answer
31 views

Minimal projective presentations from projective presentations.

Let $M$ lie in $\operatorname{mod}\Lambda$ for a finite dimensional $k$-algebra $\Lambda$. Let $P = (P_{1} \xrightarrow{d} P_{0})$ be a silting complex in $K^{b}(\operatorname{proj}\Lambda)$ and $M =...
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0answers
31 views

Inner product structure on geometric algebra?

I understand that geometric algebra equips itself with the contraction operators $\rfloor$ and $\lfloor$. While these are awesome when one wishes to project a subspace onto another, it is not an inner ...
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0answers
35 views

Reducibility Unnecessary Hypothesis?

Problem: Let $F$ be a field with $\text{char} F = p$ for some prime $p$. Show that if $X^p - X - a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$ Solution in back of the ...
2
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1answer
28 views

$M$ is cyclic if and only if there is some ideal $I\subset R$ such that $R/I\cong M$

This is the first part of exercise 11 on page 168 of Advanced linear algebra third edition of Steven Roman Let $M$ an $R$-module, then prove that $M$ is cyclic if and only if there is some ideal $I\...
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1answer
47 views

Canonical ring structure on the tensor product $R \otimes_\mathbf{Z} S$.

Let $R$ and $S$ be commutative rings. I need to show that there is a unique ring structure on the $\mathbf{Z}$-module $T := R \otimes_\mathbf{Z} S$ such that $$ (r_1 \otimes s_1)(r_2 \otimes s_2) = ...
3
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1answer
31 views

$\mathbb{Q}^{alg}[[a,b]] $ is not elementary equivalent to $\mathbb{C}[[a,b]]$, and the same for $\mathbb{Q}^{alg}[a]$ and $\mathbb{C}[a]$?

Since ACF is complete, $\mathbb{Q}^{\text{alg}}$ is elementary equivalent to $\mathbb{C}$, and by Ax-Kochen $\mathbb{Q}^{\text{alg}}[[a]]$ is elementary equivalent to $\mathbb{C}[[a]]$. But how should ...
3
votes
2answers
17 views

Let $R$ be a ring with $1$. If there exist disjoint comaximal ideals $I, J$, then for any $R$-module $M$, $M = IM \oplus JM$.

This questions is from a past year paper. Let $R$ be a ring with $1$. Suppose there exist distinct ideals $I, J$ of $R$ such that $R = I + J$ and $I \cap J = \{ 0 \}$. (a) Show that there exists $a \...
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0answers
18 views

A set which is closed under addition and is well-ordered

If I have a set of objects which support + and <, is that necessarily isomorphic to a subset of the real numbers? I ask this question because I'm trying to figure out the most general type of a ...
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1answer
24 views

Show that if N, K are normal subgroups of a group G, and N contains K then we have: $ G / N \cong (G/K) / (N /K) $ [duplicate]

Show that if $N, K$ are normal subgroups of a group $G$, and $N$ contains $K$ then we have: $$ G / N \cong (G/K) / (N /K) $$ Intuitively it looks correct, would like to know how I can approach this.
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0answers
44 views

Finding the generator of a cyclic group

Q) Show that $x$ or $2x$ is a generator of the cyclic group $(\mathbb{Z}_3[x]/\langle f(x)\rangle)^*$ where $f(x)$ is a cubic irreducible polynomial over $\mathbb{Z}_3$. My attempt: Let $F= \mathbb{Z}...
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0answers
12 views

Does ''direct sum '' in the definition of graded ring mean internal direct sum?

I just read the definition of graded ring on wiki. a graded ring is a ring that is a direct sum of abelian groups $R_{i}$such that $ R_{i}R_{j}\subseteq R_{i+j}$. Does it mean internal direct sum here?...
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1answer
76 views

Multiplication Table for $\Bbb Z_2[x] /<x^3 +x^2+x+1>$

The factor ring can be rewritten as $ \{a_o +a_1x+a_2x^2 \mid a_0,a_1, a_2 \in \mathbb{Z}_2 \}$. We figured out that the values on the multiplication table will be $\{0,1,1+x,1+x^2,x+x^2,x^2,1+x+x^2 \}...
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0answers
38 views

describe $\langle x^2, x+2\rangle$ as a subgroup

well, I totally have no idea what's the "describe" mean? the question is "describe $\langle x^2, x+2\rangle$ as a subgroup and an ideal of $\mathbb Z_4[x]$. anyone can describe the describe for me?
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1answer
26 views

Proper Ideals with Norm Relatively Prime to Conductor

Let $K$ be an imaginary quadratic number field, and $\mathcal{O}_K$ the ring of integers. Let $\mathcal{O}$ be an order. Call the $\textit{conductor}$ $f = [\mathcal{O}_K:\mathcal{O}]$. Given some $\...
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4answers
62 views

Number Theory theorem regarding ring of integers in $\mathbb{Q}[\sqrt{D}]$

Here is the theorem that I need to prove For $K = \mathbb{Q}[\sqrt{D}]$ we have $$\begin{align}O_K = \begin{cases} \mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\ \mathbb{Z}\...
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2answers
116 views

Construct a nonabelian group of order 44

Let $G$ be a group s.t. $|G|=44=2^211$. Using Sylow's Theorems, I have deduced that there is a unique Sylow $11$-subgroup of $G$; we shall call it $R$. Let $P$ be a Sylow $2$-subgroup of $G$. Then we ...
3
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2answers
75 views

''Homomorphism'' of rings

Let $R$ and $R'$ rings, $\phi$ a homomorphism of $R$ to $R'$ such that $\phi(x+y)=\phi(x)+\phi(y), \quad \forall x,y \in R$. $\phi(xy)=\phi(x)\phi(y) \quad or \quad \phi(xy)=\phi(y)\phi(x) \quad \...
1
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1answer
32 views

Show that an infinite abelian group all of whose proper quotients are finite $\cong \mathbb{Z}$

Show that an Infinite abelian group all of whose proper quoteints are finite $\cong \mathbb{Z}$ So I'm a little confused on how to get started here. Perhaps something like: Let $x \in G$, then $|G/&...
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0answers
29 views

Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble.

Let $G$ be a group s.t. $|G|=pqr$ where $p,q,r$ are primes that need not be distinct. Prove that $G$ is soluble. So, I don't know whether I should handle this case by case and try to get the Sylow ...
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votes
2answers
35 views

Corresponding space to a irreducible part of a given representation of $S_n$

My setting is the following: I have an action of $S_n$ on some $\mathbb{C}$-vector space, $V$, by permuting a special basis: $$<a^1_{1},\dots,a^1_{n!},a^2_{1},\dots,a^2_{n!},\dots,a^k_{n!},b_1,\...
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votes
0answers
16 views

Galois group as a subgroup of $S_4$ relation to constructible algebraic element

I have that $\gamma$ is a constructible element of degree $4$ over $\mathbb{Q}$, ie. $[\mathbb{Q}(\gamma):\mathbb{Q}] = 4$. Let $N$ be the normal closure of $\mathbb{Q}(\gamma)$ over $\mathbb{Q}$. The ...
0
votes
1answer
29 views

A linear transform mapping one billinear map to another may not exist (counter-example)

$$\newcommand{\im}{\mathrm{im}\;}\newcommand{\Span}{\mathrm{Span}}\newcommand{\rank}{\mathrm{rank}\;}$$For a bilinear map $\phi : V \times W \to E$ define the first nullspace as $$ N_1(\phi) = \{ v \...
1
vote
2answers
42 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
0
votes
1answer
40 views

Is there a general and mechanical method to solve algebra of sets (or alg.of propositions) equations?

In some simple cases it seems possible to solve for X a set equation. For example, if I am given : X Inter U = U , and knowing the law according to which S Inter U = S for any set S, I can find ...
0
votes
0answers
10 views

skew-symmetric non-degenerate bilinear space has even dimension [duplicate]

How to prove skew-symmetric non-degenerate bilinear space has even dimension with skew-symmetric defined as $(x,y) = - (y,x), \forall x,y \in V$, where $V$ is the vector space