-1
votes
0answers
22 views

Let $\,E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. …

Let $E = C([0, 1])$. Set $\,X =(E,||.||_\infty)$ and $\,Y =(E,||.||_1)$. Let us consider the identity $I :X→Y$. Prove that I is continuous and bijective. Calculate $\,||I||$. Prove that $I^{-1}$ is ...
0
votes
1answer
21 views

Lie bracket of vector fields on $R^2$

Compute the Lie bracket$$\Big[-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\frac{\partial}{\partial x}\Big]$$ on $R^2$ Can you help me please?
1
vote
2answers
63 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
1
vote
3answers
27 views

Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
0
votes
0answers
14 views

Finding characteristic roots and characteristic vectors

V is a two-dimensional vector space over the field of real numbers, with a basis $v_1, v_2$. Find the characteristic roots and corresponding characteristic vectors for T defined by $v_1(T) = v_1 + ...
0
votes
1answer
58 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
2
votes
1answer
21 views

Counterexample to “A/I is Artinian, when I is the annihilator of Artinian A-module”.

Let M be an Artinian A-module and let I be the annihilator of M in A. Is A/I necessarily an Artinian ring? I believe the answer is no since this comes off of a similar result regarding Noetherian ...
1
vote
1answer
26 views

Show that if $N$ is a normal subgroup of $G$ which contains all commuters then $G/N$ is abelian.

I am working on my proof for class and I was wondering if this look ok? Let $N$ be a normal subgroup of $G$ we want to show that $G/N$ is abelian, or $(aN)(aN) = abN = baN = (bN)(aN)$. Since $N$ ...
4
votes
2answers
90 views

Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$?

I have a homework question from Artin's Algebra that asks Is $i \in \mathbb{Q}[\sqrt[4]{-2}]$? I suspect that this is not true because $i \sqrt{2} \in \mathbb{Q}[\sqrt[4]{-2}]$ and $\sqrt{2}$ is ...
1
vote
1answer
35 views

Isomorphism preserves exactness

Let $R$ be a commutative ring with unity. Let $A_i$ be an R-module for every $i$. Consider a sequence of modules $$\xrightarrow{\delta_{i-1}}A_{i-1}\xrightarrow{\delta_{i}} ...
0
votes
0answers
25 views

Parity check matrix operations

Let $C_1$ and $C_2$ be linear codes of the same length over the finite field $F$, and let $H_1$ and $H_2$ be parity-check matrices of $C_1$ and $C_2$ respectively. Define $C_3$ as the code $C_3 = ...
3
votes
1answer
42 views

exercise of Matsumura

my question is about this exercise of Matsumura: in the proof hint we use is this obvious? or e should define an isomorphism?
0
votes
0answers
44 views

$\mathrm{SL}_2$ acts naturally on binary forms

Problem: Let $F^{(n)}$ be the vector space of binary forms of degree $n$ in two variables with coefficients in $\mathbb{C}$. Show that $\mathrm{SL}_2$ acts on $F^{(n)}$ and ...
1
vote
4answers
42 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
5
votes
1answer
141 views

The field of algebraic numbers in $\mathbb Q (a_1,\ldots, a_l)$ is finite over $\mathbb Q$

In the book 'Algebra IV: Infinite Groups, Linear Groups' by Kostrikin and Shafarevich, there is a sketch of a proof (on page 84) of a theorem by Schur. I'm struggling to understand the line: Since ...
1
vote
2answers
39 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
2
votes
2answers
43 views

For a shift matrix $A$, prove that $A^n=0$ but $A^{n-1} \neq 0$.

Let $A\in F_n$ be the matrix $\begin{pmatrix} 0&1&0&0&\cdots&0 \\ 0&0&1&0&\cdots&0 \\ \vdots\\ 0&0&0&0&\cdots&0 \end{pmatrix}$, whose ...
0
votes
1answer
34 views

Nilpotent group and center

Let $G$ a nilpotent group. Show that (i) If $N$ is normal to $G$ and $N \neq 1$, then $N \cap Z(G) \neq 1$ (ii) If $N$ is normal to $G$ with $|N|=p$, where $p$ is a prime number, then $N \le Z(G)$. ...
1
vote
1answer
40 views

Prove that the Polynomial Ring $F[t]$ Is an Integral Domain

Let $F$ be a field and $F[t]$ be the ring of polynomials. Take $p(t)$, $q(t) \in F[t]$, and write $p(t) = \sum_{i=1}^{n} {a_i}{t^i}$, $q(t) = \sum_{k=1}^{n} {b_k}{t^k}$, where $n$ denotes the greater ...
0
votes
0answers
33 views

Classifying quadratic extensions of $\mathbb{Q}$

I'm studying Artin's Algebra, and the question says to "Classify quadratic extensions of $\mathbb Q$." What would that look like? A quadratic extension of $\mathbb Q$ is just, for $d$ square-free, ...
0
votes
1answer
24 views

Show that the equivalence classes of $\sim$ are left cosets of $H$ in $G$.

Let $H \leq G$ and define a relation on $G$ by $x \sim y$ if $y^{-1}x \in H$. Show that $\sim$ is an equivalence relation on $G$ and then show that the equivalence classes of $\sim$ are left cosets ...
0
votes
1answer
35 views

Show that $T(X) =\{f\in A(S)|f(X)\subset X\}$ is a subgroup of $A(S)$ if X is finite.

If $S$ is a nonempty set and $X\subset S$, Show that $T(X)=\{f\in A(S)|f(X)\subset X\}$ is a subgroup of $A(S)$ if X is finite. Note: $A(S)$ is called "symmetric group". It's actually a collection of ...
2
votes
3answers
60 views

The meaning of $\bigcap_{x\in G} x^{-1}Hx$ and the proof for the fact that $N$ is a subgroup of $G$ such that $y^{-1}Ny=N$ for every $y\in G$

If $H$ is a subgroup of $G$, let $N=\bigcap_{x\in G}x^{-1}Hx$. Prove that $N$ is a subgroup of $G$ such that $y^{-1}Ny=N$ for every $y\in G$. What does $N=\bigcap_{x\in G}x^{-1}Hx$ mean? I'm confused ...
0
votes
2answers
51 views

Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.

If $A$ and $B$ are finite subgroups, of orders $m$ and $n$, respectively, of the abelian group $G$, Prove that $AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime. Definition: ...
1
vote
1answer
25 views

Let E be defined over Fq and let n ≥ 1. Show that E(Fq)[n] and E(Fq)/nE(Fq) have the same order.

Let E be an Elliptic Curve defined over $F_q$ and let n ≥ 1. Show that $E(F_q)[n]$ and $E(F_q)$/$nE(F_q)$ have the same order. I feel like this is obvious. The n-th torsion group $E(F_q)[n]$ ...
3
votes
2answers
29 views

Prove that $\lambda_a$ is a permutation of a group $G$ for a fixed element $a \in G$.

Hi I am working on following hw problem and I want to make sure that I am doing this correctly? I think I am going about this in the right way but I still need some reassurance. Let $G$ be a group ...
0
votes
5answers
105 views

Is this a true theorem? [closed]

I'm trying to prove the existence of the following theorem: If $n,p \in \mathbb{N}$, then $(p+1)^n = 1 \mod p$ Is this theorem true? I think it is, but I don't know how to prove it! Thanks!
1
vote
2answers
38 views

How can you find all subgroups of a certain group?

For example, how can how find all subgroups of $S_3$? How can you ensure that your answer includes all of them? Note: I want to know if there is a universal method to find all subgroups for a single ...
2
votes
2answers
43 views

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.

$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of $F$ have a square root. (Hint: For some integer $i$, ($1\leq i \leq s-1$), $i$ is odd if and only if ...
1
vote
2answers
39 views

Given a finite abelian group $G$ with $g \in G$, then for any divisor $d$ of $|g|$ there is an element of $G$ with order $d$.

From an homework question that comes as an introduction to abelian groups. Regarding my efforts to solve the question, I have been trying to utilize the fundamental theorem of finite abelian groups, ...
0
votes
1answer
45 views

Set of elements of order 2 in a group.

For a group G, define $G_2$={ $g\in G$:$|g|=2$ }. Prove that if $G_2$ is finite, then $|G_2|$ is odd. Can you help me please?
3
votes
1answer
48 views

Simpler method to show $k(x)$ is NOT a primitive irreducible polynomial?

Let $k(x) = x^2+2x+2$ be under $\mathbb Z_{11}[x]$. Determine if $k(x)$ is irreducible, and if so, determine if it is primitive. Ok, I showed that $k(x)$ is irreducible since it it a quadratic and ...
0
votes
1answer
33 views

With B integral over subring A, homomorphism from A to algebraically closed field F can be extended to B.

Here's the problem I am working on: Let A be a subring of B such that B is integral over A, and let $f: A \rightarrow F$ be a homomorphism of A into an algebraically closed field F. Show that f ...
-1
votes
0answers
35 views

Subgroups with 300 elements [duplicate]

Show that every group which order 300 have a normal subgroup. I try show this for contradiction with Sylow's Theorem but I don't know what I have to do before that...
0
votes
1answer
27 views

Radical of an ideal - prove every prime ideal that contains $I$ also contains $\sqrt{I}$

Let $R$ be a commutative ring with a unit and $I$ an ideal. Please prove that every prime ideal that contains $I$ also contains $\sqrt{I}$. I easily conclud that $I \subseteq \sqrt I$ but I ...
0
votes
3answers
82 views

showing $Q[\sqrt 2] = Q(\sqrt 2)$

The question came in my exam. $Q[\sqrt 2] = \{ a + b \sqrt2 \;| a,b \in Q\}$ and $Q(\sqrt 2)$ is minimal subfield of it's extension containing $Q$ and $\sqrt 2$. (In my book) It calls $F(a)$ ...
0
votes
5answers
33 views

Trying to prove that if $\sigma$ and $\tau$ are disjoint cycles and if $\sigma\tau = id$ then $\sigma = id$ and $\tau = id$

Let $\sigma$ and $\tau$ be two disjoint cycles in $S_n$, if $\sigma\tau = id $ then, $$\sigma\tau(id) = \sigma(\tau(id)) = \sigma(id) = id$$ Concluding that both $\sigma = id$ and $\tau = id$. ...
2
votes
0answers
44 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
1
vote
1answer
29 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
4
votes
0answers
35 views

Unicity inner automorphism symmetric group

I need to prove that the center of the symmetric group $S_n$ with $n\geq 3$ is trivial. I chose to use Lagrange's theorem: bearing in mind that the center is a subgroup, it is easily found that ...
2
votes
2answers
54 views

“Primeness” of C[x] in B[x], where A is a subring of B and C is the integral closure of A in B.

Let A be a subring of B, and C the integral closure of A in B. If f, g are monic polynomials in B[x] such that fg is in C[x], then f, g are in C[x]. The first part of the problem allowed the ...
-1
votes
0answers
59 views

cyclic group and abelian

this is my question and i have produced some solution. i do not know whether it is the correct one or not. Let G_1,….,G_n be groups. Prove that the direct product G_1x….xG_n is abelian if and ...
1
vote
0answers
33 views

Ring A is integral over the subring of invariants under a finite group action

I need to prove that if G is a finite group that acts on ring A, and $A^G$ is the subring consisting of elements of A which are invariant under all g in G, then A is integral over $A^G$. The hint ...
3
votes
3answers
78 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
1
vote
1answer
36 views

Normal subgroup, and cosets.

problem: For a subgroup $N$ of a group $G$, prove if $N$ is normal, then each left coset of $N$ is also a right coset, that is for all $a \in G$, there exists a $b \in G$, such that $aN = Nb$. ...
1
vote
1answer
20 views

Prove that any conjugate of $a$ has the same order as $a$.

I am trying to prove that any conjugate of $a$ has the same order as $a$. Let $G$ be a group and let $a \in G$. An element $b \in G$ is called a conjugate of $a$ if $b=xax^{-1}$. My professor gave ...
0
votes
1answer
37 views

finitely generated right R-module which is not cyclic

Let $A$ be a finitely generated right $R$-module which is not cyclic. Prove that there exist $B \le A_R$ maximal with respect to the property that $A/B$ is not cyclic. help please and thank you for ...
0
votes
3answers
59 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
2
votes
1answer
39 views

If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

Let $A$ be a subring of ring $B$, with $B$ integral over $A$. If $x$ in $A$ is a unit in $B$, then it is a unit in $A$. I know that $f(t) = t - x$ is in $A[t]$ with $f(x) = 0$, and that there ...
2
votes
2answers
38 views

Trying to determine if $G = \mathbb{R}$ then $H = \{\log a \mid a \in \mathbb{Q}, a > 0\}$ is a subgroup.

I am trying to prove that if $G = \mathbb{R}$ then $H = \{\log a \mid a \in \mathbb{Q}, a > 0\}$ is a subgroup. The identity of $G$ is $0$ and $0 \in H$. If $a,b \in \mathbb{Q}$ and $a>0$ and ...