For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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8
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213 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 ...
7
votes
0answers
77 views

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
7
votes
0answers
284 views

Fixed point of a shrinking map. Proof.

Could you please verify my proof? Let $f:X \to X$ be a shrinking map on a compact metric space $(X,d)$. In other words: $d(f(x),f(y)) < d(x,y)$ for all $x,y \in X$. Prove: $f$ has a unique ...
6
votes
0answers
51 views

$E \to S$ surjective in degrees $\geq 1$ implies $\widetilde{E} \to \widetilde{S}$ surjective

In the proof of Theorem II.8.13 in Hartshorne (which is the Euler sequence), the author writes: Let $S = A[x_0, \ldots, x_n]$. [...] The exact sequence $$0 \to M \to E \to S$$ of graded ...
6
votes
0answers
126 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
6
votes
0answers
115 views

Proposed proof of analysis result

Hi please advise on my proof of the following result: Assume that $I \subset \mathbb{R}^{n}$ is convex, bounded open set with Lipschitz boundary and let $u_{m},u$ be such that $$u_{m} ...
6
votes
0answers
94 views

A power of the characteristic polynomial

Let $A$ be a square matrix with real or complex coefficients of size $n$. Define its characteristic polynomial by $\chi_A(X) = \det(A-XI_n)$ (or $\det(XI_n-A)$ if you prefer). The question is : Prove ...
6
votes
0answers
94 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
6
votes
0answers
185 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
6
votes
0answers
379 views

bisectors of an angle in a triangle intersect at a single point - proof verification

Let´s consider a general triangle ABC. Let´s draw two angle bisectors from vertices A and B. It is obvious that these two angle bisectors intersect at a single point X. Since X lies on the angle ...
6
votes
0answers
649 views

Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$

As a homework question, we got asked the following: Construct a function $f:[0,1] \rightarrow \mathbb{R}$ which is not the pointwise limit of any sequence of continuous functions Thinking about ...
6
votes
0answers
128 views

Proof of inverse function theorem by approximation property

In proving the inverse function theorem using the approximation characterization of the derivative, we are given $F:\mathbb{R}^n \to \mathbb{R}^n$ such that  $$F(p_0 + h) - F(p_0) = DF_{p_0}(h) + ...
6
votes
0answers
219 views

Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb ...
5
votes
0answers
73 views

In which of the three topologies is $f(t)=(t, 2t, 3t, 4t, \ldots)$ continuous? Here, $f$ is a function from $\mathbb{R}$ to $\mathbb{R}^\omega$.

Can someone please verify my proof or offer suggestions for improvement? Consider the product, uniform, and box topologies on $\mathbb{R}^\omega$. In which of the three topologies is $f(t)=(t, 2t, ...
5
votes
0answers
95 views

Jech, “Set theory” exercises 12.11 - Is my proof right?

I try to prove the Jech's "Set theory", exercises 12.11: 12.11. If $\kappa$ is an inaccessible cardinal, then $V_\kappa\models \text{there is a countable model of ZFC}$. My attempt. Since ...
5
votes
0answers
125 views

Complex Conjugation as an $F$-Automorphism of $K$?

I am struggling with the following problem: Let $f \in F[x]$ be an irreducible quintic polynomial with splitting field $K$, where $\mathbb{Q} \subseteq F$. Supposing that $f$ has three real roots ...
5
votes
0answers
109 views

$f$ continuous nowhere implies $f$ not Riemann integrable

I am trying to prove the statement in the title, without using the theorem which says that the set of discontinuities of a Riemann integrable function must have measure $0$. I am aware that similar ...
5
votes
0answers
81 views

representation theorem on the path space

I'm working on a project and have done some work. However, there are some point where I'm unsure if my thoughts are correct. It would be appreciated if someone could share their thoughts about it. ...
5
votes
0answers
229 views

Prove that $\mathbb{R}^k$ is separable

I'd like to show that $\mathbb{R}^k$ is separable. (A metric space is called separable if it contains a countable dense subset.) Here's what I have and I'd like to confirm with everyone to see if ...
4
votes
0answers
36 views

Limit of continuous function

Prove or provide a counterexample: 1) $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. If $(a_{n}) = f(n)$ converges to $L$, then $\lim_{x \rightarrow \infty} f(x) = L$. Counterexample: I ...
4
votes
0answers
36 views

My proof of: Every convergent real sequence is a Cauchy sequence.

Is my proof correct? Let $(x_n)_{ n \in \mathbb{N} }$ be a real sequence. $\textbf{Definition 1.}$ $(x_n)$ is $\textit{convergent}$ iff there is an $x \in \mathbb{R}$ such that, for every ...
4
votes
0answers
51 views

Check proof about range of bounded linear operator.

I have to prove that the range $\mathcal{R}(T)$ of bounded linear operator $T:X\rightarrow Y$; $X,Y$ normed spaces need not be closed in $Y$. As a hint I'm given that I could consider ...
4
votes
0answers
38 views

Every $\sigma-$finite measure is semifinite. $(X, \mathcal{M}, \mu)$ is a measure space.

Definition 1: Say $X = \bigcup_{n=1}^{\infty} E_n $ where $E_n \in \mathcal{M}$ and $\mu( E_n ) < \infty $ for all $n$, we call $\mu$ $\sigma$-finite. More generally, if $E = \bigcup^{\infty} E_n ...
4
votes
0answers
62 views

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification)

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification) Proof: Let G be generated by c, b, where $c^2 = b^2 = 1$. ...
4
votes
0answers
134 views

6-digit password - a special decoding method

Consider the situation of decoding a 6-digit password that consists of the symbols A to Z and 0 to 9, where all possible combinations are tried randomly and uniformly. Consider the ...
4
votes
0answers
62 views

$\mu$-clubs and stationary sets consisting of elements with cofinality $\mu$

Let $\mu < \kappa$ be infinite cardinals. A set $C$ is called a $\mu$-club in $\kappa$, if it is unbounded in $\kappa$ and contains all its limit points of cofinality $\mu$. Now let $T \subset S ...
4
votes
0answers
91 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
4
votes
0answers
77 views

$\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0}\cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$

I'd like someone to check my proof that $\aleph_\alpha^{\aleph_1} = \aleph_\alpha^{\aleph_0} \cdot 2^{\aleph_1}$ for all $\omega \le \alpha < \omega_1$. First, let's prove that ...
4
votes
0answers
73 views

Two question about how to compute this integral limit

Let $f: (-\pi,\pi]\to \mathbb R$ be continuous and let $p_x (u) = {(f(u+x) - f(x)) \cos ({u \over 2}) \over \sin ({u \over 2}) }$. I want to show that $$ \int_{-\pi}^\pi p_x(u) \sin (Nu) du \to 0$$ ...
4
votes
0answers
72 views

If $f_n(t):=f(t^n)$ converges uniformly to continuous function then $f$ constant

Let $f \colon [0,1] \to \mathbb R$ be continuous and let $f_n$ be defined by $$ f_n(x):=f(x^n), \qquad x \in [0,1], \,\, n \in \mathbb N. $$ Suppose there exists a continuous function $g$ on ...
4
votes
0answers
87 views

Proof “correctness” : Cycle structure of conjugate permutations

My Algebra lecturer is a very strict about proofs(w.r.t Completeness , correctness and format ) more so than I have encountered in the past or any of my lecturers of the courses I am take concurrent. ...
4
votes
0answers
63 views

Unique representation of a degenerate simplex

I recently learned about simplicial sets, and now I'm studying some basic properties. I've learned that every degenerate simplex is a degeneracy of a unique non-degenerate (nice) simplex. However, I ...
4
votes
0answers
24 views

Integration relating to transported measures

Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}')$ two measurable spaces and $T\colon (\Omega,\mathcal{A})\to (\Omega',\mathcal{A}')$ measurable. Show: If $f\colon ...
4
votes
0answers
552 views

Prove that every separable metric space has a countable base.

A collection $\{V_\alpha \}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x \in X$ and every open set $G \subset X$ such that $x \in G$, we have $x \in ...
4
votes
0answers
55 views

Nontrivial homomorphism for $Z_a \times Z_b $to $Z_c \times Z_d$ - Fraleigh p. 134 13.35

This isn't a duplicate of this. Let $(A, B) \in \mathbb{Z_a \times Z_b}$. Hinging on p. 2, I guess homomorphism is $h(A,B) = (A \text{ mod } c, B \text{ mod } d)$. I'm unsettled. p. 2 sprang it up ...
4
votes
0answers
437 views

Finding the Green Function of the upper half ball

Find the Green function of $\Omega:=\left\{x\in\mathbb{R}^n:\lVert x\rVert<R, x_n>0\right\}$ and show that the function you've found is indeed a Green function! You are allowed to use ...
4
votes
0answers
91 views

Order of the Normalizer of a Sylow $p$-subgroup in $S_{p}$

Question: Let $p$ be a prime and $P \leq S_{p}$ with $|P| = p$. Prove that $|N_{S_{p}}(P)| = p(p-1)$. I have already solved this problem, but I have come across a proof that is much more elegant than ...
4
votes
0answers
68 views

Is this proof that $|S_n|=n!$ correct?

I've always heard that it's trivial that $|S_n|=n!$ where $S_n$ is the symmetric group of degree $n$. Now, my proof was the following: consider $I_n=\{1,\dots,n\}$ and consider ...
4
votes
0answers
41 views

General and basic question about convergence of a series

Let $(a_{i,j})_{i,j=1}^n$ be a sequence of real numbers such that the following series converges $$ S = \lim_{n\to\infty}\sum_{i=1}^n\sum_{j=1}^na_{i,j} $$ It is known that for each $i$th the ...
4
votes
0answers
140 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n ...
3
votes
0answers
44 views

Prove the function is integrable

For a point $x \in [1,2]$, define $f(x) = 0$ if $x$ is irrational and define $f(x)= \frac 1n$ if $x$ is rational and is expressed as $x = \frac mn$ for natural numbers $m$ & $n$ having no common ...
3
votes
0answers
20 views

Proof Verification: Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$.

Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$. Proof: First, prove that S is a subgroup of R. Pick an arbitrary element $x$ from ...
3
votes
0answers
35 views

Verify my induction proof for Balanced Ternary expressions

In first place I apologize if you find a grammatical error, my English is not too good for now, but I'm work on it. That also goes for errors in my question (this is my first post). I encounter the ...
3
votes
0answers
23 views

TRying to show the following collection is a sigma algebra

Let $f: \Omega \to E $ be a function and let $\mathscr{E}$ be a sigma algebra of subsets of $E$. Let $$ \mathscr{A} = \{ A \subset \Omega : \; \text{there exists} \; B \in \mathscr{E} \; \text{with} ...
3
votes
0answers
42 views

if $X\subseteq Y$, then $\bar{X} \subseteq \bar{Y}$

Let $X$, $Y$ be arbitrary subsets of $\mathbb{R}$, show if $X\subseteq Y$, then $\bar{X} \subseteq \bar{Y}$ Proof since $Y\subseteq \bar{Y}$ $\bar{Y}$ contains all the adherent points of Y by ...
3
votes
0answers
53 views

Deriving a formula for an initial boundary-value problem

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
3
votes
0answers
58 views

Problem on symmetric polynomials

The following problem is from "Analysis I" by Amann/Escher. Exercise: There are obvious operations of $S_m$ on $\mathbb{N}^m$ and on $R[X_1,\dots,X_m]$. A polynomial $p\in R[X_1,\dots,X_m]$ is called ...
3
votes
0answers
49 views

Local Artinian rings with a principal maximal ideal

I would be very grateful if someone would check my proof of the following result (this is not homework). All rings are commutative and unital. Proposition: If $(A,\mathfrak{m})$ is a local Artinian ...
3
votes
0answers
33 views

Showing points of continuity of a function f(x) that takes the value 1/n whenever x belong to a sequence {An} and is zero elsewhere.

I am given a sequence $(An), n=1,2,3,...$ which consists of distinct numbers, which converges to $3$ as $n$ tends to infinity, but none of its terms are equal to $3$. Then I am given a function $f(x) ...
3
votes
0answers
27 views

On the greatest lower bound property

Proposition: Let $S$ be an ordered field and $S \supset E \neq \varnothing $. $E$ is bounded below. Then $ \inf E = - \sup ( - E ) $ Try: Write $- E = \{ -x : x \in E \} $ and let $l $ be a lower ...