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  1. F

    Finding the dimension of a subspace

    I am stuck on finding the dimension of the subspace. Here's what I have so far. Proof: Let ##W = \lbrace x \in V : [x, y] = 0\rbrace##. We see ##[0, y] = 0##, so ##W## is non empty. Let ##u, v \in W## and ##\alpha, \beta## be scalars. Then ##[\alpha u + \beta v, y] = \alpha [u, y] + \beta [v...
  2. F

    Showing two groups are equal/Pontryagin duality

    I am confused because ##H## is a subgroup of ##G## and ##H^{\perp\perp}## is a set of homomorphisms. Are we trying to show ##f(H) = H^{\perp\perp}## where ##f## is the isomorphism defined in the definitions above? Proof: We want to show ##H = (H^\perp)^\perp##. ##(\subset):## Let ##h \in H##...
  3. F

    Lower central series

    My attempt: If ##i = 1##, then ##\gamma_1 = G \rhd G' = \gamma_2##. We proceed by induction on ##i##. Consider an element ##xyx^{-1}y^{-1}## where ##x \in \gamma_i## and ##y \in G##. Since ##\gamma_i \rhd G##, we have ##yx^{-1}y^{-1} = x_0 \in \gamma_i##. So, ##xyx^{-1}y^{-1} = xx_0 \in \gamma_i...
  4. F

    Refining a normal series into a composition series

    Attempt: Consider an arbitrary normal series ##G = G_0 \ge G_1 \ge G_2 \ge \dots \ge G_n = 1##. We will refine this series into a composition series. We start by adding maximal normal subgroups in between ##G_0## and ##G_1##. If ##G_0/G_1## is simple, then we don't have to do anything. Choose...
  5. F

    Normal series and composition series

    Attempt so far: We're given that ##G## and ##H## have equivalent normal series $$G = G_0 \ge G_1 \ge \dots \ge G_n = 1$$ and $$H = H_0 \ge H_1 \ge \dots \ge H_n = 1$$ We can assume they have the same length because they are equivalent. I think from here I need to construct two composition...
  6. F

    Normal group of order 60 isomorphic to A_5

    Proof: We note ##60 = 2^2\cdot3\cdot5##. By Sylow's theorem, ##n_5 = 1## or ##6##. Since ##G## is simple, we have ##n_5 = 6##. By Sylow's theorem, ##n_3 = 1, 4, ## or ##10##. Since ##G## is simple, ##n_3 \neq 1##. Let ##H## be a Sylow ##3## subgroup and suppose ##n_3 = 4##. Then ##[N_G(H) : G] =...
  7. F

    Prove Frattini's argument

    Attempt at solution: Proof of i): Let ##x \in X##. Its clear ##G \supseteq HG_x##. Let ##g \in G##.Then there is ##y \in X## such that ##g \cdot x = y##. Since ##H## acts transitively on ##X##, there is ##h \in H## such that ##h \cdot x = y##. So, ##g \cdot x = h \cdot x##. This gives...
  8. F

    Dim null ST <= dim null S + dim null T

    Proof: Since ##U, V## are finite dimensional, we have that ##\operatorname{null} S, \operatorname{null} T## are finite dimensional. Let ##v_1, \dots v_m## be a basis of ##\operatorname{null}S## and ##u_1, \dots, u_n## be a basis of ##\operatorname{null} T##. It is enough to show there are ##m +...
  9. F

    Which sigma algebra is this function a measure of?

    Suppose ##\nu## is a measure on some ##\sigma##-algebra ##\mathcal{A}##. Then we must have for all ##A \in \mathcal{A}## either ##A## or ##A^c## is finite, but not both. Because otherwise ##\nu(A)## is undefined or not well defined. I've verified that ##\lbrace \emptyset, X \rbrace## and...
  10. F

    Extending a pre measure

    Attempt at solution: We have ##\sum = \lbrace B \subset \mathbb{R} : b \in B \Rightarrow -b \in B \rbrace##. Clearly, ##\emptyset \in \sum##. Let ##B \in \sum## and ##a \in B^c##. Then ##a \not\in B## which implies ##-a\not\in B##. So ##-a \in B^c## i.e. ##B^c \in \sum##. Lastly, for any...
  11. F

    Monotone class / Borel sets

    Proof: Let ##A, B \in \mathcal{O}## and ##x \in A \cap B##. Then there exists ##\varepsilon_A, \varepsilon_B > 0## such that ##B_{\varepsilon_A}(x) \subset A## and ##B_{\varepsilon_B}(x) \subset B##. Let ##\varepsilon = \min\lbrace\varepsilon_A, \varepsilon_B\rbrace##. Then ##B_\varepsilon(x)...
  12. F

    Monotone classes proof

    My question is how to show ##\mathcal{F} \subset \sum'##. Here is my work for the problem: Proof of hint: First we'll show ##\sum## is a monotone class. Let ##(A_n)_{n\in\mathbb{N}} \subset \sum## and ##F \in \mathcal{F}##. There are two things to verify. Suppose ##(A_n) \uparrow A =...
  13. F

    Infinite union of sigma algebras

    For all ##n\in\mathbb{N}## we have ##\emptyset \in A_n##. Hence, ##\emptyset \in \mathcal{A}_\infty##. Let ##A \in \mathcal{A}_\infty##. Then ##A \in A_k## for some ##k\in\mathbb{N}##. So ##A^c \in A_k##. Hence, ##A^c \in \mathcal{A}_\infty##. Thus, ##\mathcal{A}_\infty## is closed under...
  14. F

    Field with char p

    Suppose ##f## is reducible over ##F##. Then there exists ##g, h \in F## such that ##g, h## are not units and ##f = gh##. If there exists ##b \in F## such that ##b^p = a##, then ##(x - b)^p = x^p - b^p = x^p - a##, using the fact that ##F## has characteristic ##p##. So, if such a ##b \in F##...
  15. F

    Polynomial ring zero divisors

    Proof: ##(\Leftarrow)## Suppose there exists non zero ##b \in R## such that ##bp(x) = 0##. Well, ##R \subset R[x]##, and so by definition of zero divisor, ##p(x)## is a zero divisor. (assuming ##p(x) \neq 0##). ##(\Rightarrow)## Suppose ##p(x)## is a zero divisor in ##R[x]##. Then we can choose...
  16. F

    F is an isomorphism from G onto itself,..., show f(x) = x^-1

    i) Proof: Let ##a, b \in G## ##(\Rightarrow)## If ##G## is abelian, then ## \begin{align*} f(a)f(b) &= a^{-1}b^{-1} \\ &= b^{-1}a^{-1} \\ &= (ab)^{-1} \\ &= f(ab) \\ \end{align*} ## So ##f## is a homomorphism. ##(\Leftarrow)## If ##f## is a homomorphism, then ## \begin{align*}...
  17. F

    Sequence of integrable functions (f_n) conv. to f

    ##\textbf{Attempt at solution}##: If I can show that ##f## is integrable on ##[a,b]##, then for the second part I get : Let ##\frac{\varepsilon}{b-a} > 0##. By definition of uniform convergence, there exists ##N = N(\varepsilon) > 0## such that for all ##x \in [a,b]## we have ##\vert f(x) -...
  18. F

    Show |x^2 sin^8(e^x)| <= (16 pi^3)/3

    ##\textbf{Attempt at solution:}## By theorem 33.5., $$\vert \int_{-2\pi}^{2\pi} x^2\sin^8(e^x) dx \vert \le \int_{-2\pi}^{2\pi} \vert x^2\sin^8(e^x) \vert dx $$ $$= \int_{-2\pi}^{2\pi} \vert x^2 \vert \vert \sin^8(e^x) \vert dx$$ $$\le \int_{-2\pi}^{2\pi} \vert x^2 \vert dx =...
  19. F

    Mean Value Theorem

    a) Proof: By theorem above, there exists a ##a \in \mathbb{R}## such that for all ##x \in I## we have ##f'(x) = a##. Let ##x, y \in I##. Then, by Mean Value Theorem, $$a = \frac{f(x) - f(y)}{x - y}$$ This can be rewritten as ##f(x) = ax - ay + f(y)##. Now, let ##g(y) = -ay + f(y)##. Then...
  20. F

    Definition of derivative

    Proof: By definition of derivative, $$f'(a) = \lim_{x\rightarrow a}\frac{f(x) - f(a)}{x - a}$$ exists and is finite. Let ##(x_n)## be any sequence that converges to ##a##. By definition of limit, we have $$\lim_{x_n\rightarrow a} \frac{f(x_n) - f(a)}{x_n - a} = f'(a)$$. By definition of...
  21. F

    Group action and blocks

    Proof: Let ##B = \lbrace a \rbrace \subseteq A## and ##\rho \in S_4##. We have two cases, ##\rho(a) = a## in which case ##\rho(B) = B##, or ##\rho(a) \neq a## in which case ##\rho(B) \cap B = \emptyset##. Its clear that ##\rho(A) = A##. So these sets are indeed blocks. Now let ##C## be any...
  22. F

    Free group definition

    Let ##S = \lbrace a, b \rbrace## and define ##F_S## to be the free group, i.e. the set of reduced words of ##\lbrace a, b \rbrace## with the operation concatenation. We then have the universal mapping property: Let ##\phi : S \rightarrow F_S## defined as ##s \mapsto s## and suppose ##\theta : S...
  23. F

    Is the set open, closed, neither, bounded, connected?

    Let ##z = a + bi##. Using the definition of modulus, we have ##\vert z - 3 \vert < 2## is equivalent to ##\sqrt{(a+3)^2 + b^2} < 2##. Squaring both sides we get ##(a+3)^2 + b^2 < 4##. This is the open disk center at ##3## with radius ##4## which we write as ##D[-3, 2]##. First we show...
  24. F

    Center of R, F in D_n

    Let ##F_0## be a reflection in ##D_n## s.t. ##F \neq F_0##. Observe, ##F_0F = FF_0## is equivalent to ##F_0FF_0F = (F_0F)^2 = R_0##. Since a reflection followed by a reflection is a rotation, and the only rotation of order 2 is ##R_{180}##, we have ##F_0F = R_{180}##. Thus, ##F_0F = FF_0## is...
  25. F

    Longest path in a connected graph

    Let ##P = (u_1, u_2, \dots, u_7)## and ##P' = (v_1, v_2, \dots, v_7)##. If there were a vertex ##w## such that ##w## is adjacent to ##u_1## and for all ##i##, ##u_i \neq w##, then we'd have a path of length 8 ##(w, u_1, u_2, \dots, u_7)##. So no such ##w## exists in ##G##. By definition of...
  26. F

    Almost irregular graph

    I did the cases ##n=2, 4, 6, 8## by hand and got this far: Answer: If ##n = 2k## for some integer ##k##, the equal degrees is ##k##. Proof: Let ##n \ge 2## be an integer and consider two cases: case1: ##n = 2k## for some integer ##k##. Let ##P(2l)## be the assertion that for ##2l \ge 2## there...
  27. F

    U(n) as an External Direct Product

    Homework Statement EDIT: ##U(n)## is the set of relatively prime numbers less than ##n## and ##U_k(n) = \lbrace x \epsilon U(n) : x \equiv 1 (\operatorname{mod} k) \rbrace##. I'm trying to finish the proof of this statement(s): Suppose ##s## and ##t## are relatively prime. Then ##U(st) \approx...
  28. F

    If a sequence is Uniformly Cauchy, that implies uniform convergence for the sequence?

    Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...
  29. F

    Integrating an exponential function

    Homework Statement Show ##\int_{0}^{1}e_n(x)\overline e_k(x) dx = 1## if ##n=k## and ##0## otherwise. Homework Equations ##e_n(x) = e^{2\pi inx}##. The Attempt at a Solution Consider 2 cases: case 1: ##n=k##. Then ##\int_{0}^{1} e_n(x) \bar e_k(x) dx = \int_{0}^{1} e_n(x)e_{-k}(x) dx =...
  30. F

    Complex numbers sequences/C is a metric space

    Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...
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