Solve your IGNOU Doubts

Question:

Let p be an odd prime that divides $n, n = p^r l, p \nmid l$ . Suppose $C = \langle x^l \rangle$ . Show that C is the unique Sylow p-subgroup of D_n.

See Answer →

Question:

In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group

$$D_n = \langle x, y : x^n, y^2, yxyx \rangle, n \in \mathbb{N}, n \ge 2.$

See Answer →

Question:

Show that
$\quad$ (i) Show that a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in GL_2(\mathbb{R})$ is symplectic if and only if $ad - bc = 1$ .
$\quad$ (ii) Show that, to prove that $SP_2(\mathbb{R})$ acts transitively on $\mathbb{R}^2 \setminus \{\mathbf{0}\}$ , it is enough to show that, for any vector $\begin{bmatrix} a \\ b \end{bmatrix} \neq \mathbf{0} \in \mathbb{R}^2$ , there is a $2 \times 2$ symplectic matrix with $\begin{bmatrix} a \\ b \end{bmatrix}$ as the first column. (Hint: For any matrix A, what is $A \begin{bmatrix} 1 \\ 0 \end{bmatrix}$ ?)
$\quad$ (iii) Complete the proof by showing that, given any non-zero vector $\begin{bmatrix} a \\ b \end{bmatrix}$ , there is always a non-zero vector $\begin{bmatrix} a' \\ b' \end{bmatrix}$ such that $\begin{bmatrix} a & a' \\ b & b' \end{bmatrix}$ is symplectic.

See Answer →

Question:

The aim of this exercise is to show that $SP_2(\mathbb{R})$ acts transitively on $\mathbb{R}^2 \setminus \{\mathbf{0}\}$

See Answer →

Question:

Suppose that $M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$ is $2n \times 2n$ matrix where A, B, C and D are $n \times n$ matrices. Show that M is symplectic if and only if the following conditions are satisfied:
$\quad$ $A^t D - C^t B = \mathbf{I}$
$\quad$ $A^t C - C^t A = \mathbf{0}$
$\quad$ $B^t D - D^t B = \mathbf{0}$
$\quad$ (Hint: Use block matrix multiplication.)
$\quad$ Also, check that the matrix $\begin{bmatrix} \mathbf{0} & -A \\ A & \mathbf{0} \end{bmatrix}$ , where A is a $n \times n$ orthogonal matrix, is a symplectic matrix.

See Answer →

Question:

If $\mathbb{F}$ is a finite field show that there is always an irreducible polynomial of the form x³ - x + a where $a \in \mathbb{F}$ . (Hint: Show that $x \mapsto x^3 - x$ is not a surjective map.)

See Answer →

Question:

By looking at the factorisation of $x^9 - x \in \mathbb{F}_3[x]$ guess the number of irreducible polynomials of degree 2 over $\mathbb{F}_3$ . Find all the irreducible polynomials of degree 2 over $\mathbb{F}_3$ .

See Answer →

Question:

If $char(F) \neq 2$ , show that a polynomial ax² + bx + c is irreducible iff $b^2-4ac\notin\mathbb{F}^{</em>2}$ where $\mathbb{F}^{<em>2}$ is the group of squares in $\mathbb{F}^{</em>}$

See Answer →

Question:

Find $[\mathbb{Q}(\sqrt[3]{2}, \omega) : \mathbb{Q}]$ where $\omega^3 = 1, \omega \neq 1$ .

See Answer →

Question:

Let $F \subset K$ and let $\alpha, \beta \in K$ be algebraic over F of degree m and n, respectively. Show that $[F(\alpha, \beta) : F] \le mn$ . What can you say about $[F(\alpha, \beta) : F]$ if m and n are coprime?

See Answer →

Question:

) Let $F(\alpha)$ be a finite extension F of odd degree (greater than 1). Show that $F(\alpha^2) = F(\alpha)$

See Answer →

Question:

Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
(b) Is there a finite group with class equation 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2?
(c) Compute the following:
$\quad$ a) $\left(\frac{173}{211}\right)$ $\quad$ b) $\left(\frac{167}{239}\right)$

See Answer →

Question:

Describe the set of primes p for which x² - 11 splits into linear factors over $\mathbb{Z}_p$ .

See Answer →

Question:

Find the elementary divisors and invariant factors of $\mathbb{Z}_8 \times \mathbb{Z}_{12} \times \mathbb{Z}_{15}$ .

See Answer →

Question:

Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that P is normal in K.

See Answer →

Question:

Check that $\mathbf{I} - \mathbf{b}$ is in the stabiliser of $\mathbf{x}$ . Also, show that there are infinitely many choices of $\alpha$ for which $\mathbf{I} - \mathbf{b}$ is invertible.

See Answer →

Question:

Let $\begin{bmatrix} a' \\ b' \end{bmatrix}$ be a vector that is not a scalar multiple of $\begin{bmatrix} a \\ b \end{bmatrix}$ . Show that there is a matrix $\mathbf{b} = \begin{bmatrix} u & v \\ w & z \end{bmatrix}$ such that $\mathbf{b}\begin{bmatrix} a \\ b \end{bmatrix} = \mathbf{0}$ and $\mathbf{b}\begin{bmatrix} a' \\ b' \end{bmatrix} = \alpha\begin{bmatrix} a' \\ b' \end{bmatrix}$ . (Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)

See Answer →

Question:

Suppose that $\det(\mathbf{x}) = 0$ in the remaining parts of this exercise. We will show that the stabiliser of $\mathbf{x}$ is infinite. If $\mathbf{x} = \mathbf{0}$ , the stabiliser of $\mathbf{x}$ is $GL_2(\mathbb{R})$ . So suppose $\mathbf{x} \neq \mathbf{0}$ . Let us write $\mathbf{x} = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$ . Then, $\begin{bmatrix} a \\ b \end{bmatrix} = \lambda \begin{bmatrix} c \\ d \end{bmatrix}$ for non-zero $\lambda \in \mathbb{R}$ . Why ?

See Answer →

Question:

Consider the natural action of $GL_2(\mathbb{R})$ on $\mathbf{M}_2(\mathbb{R})$ , the set of $2 \times 2$ real matrices, by left multiplication.

See Answer →

Question:

Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then $G_{s_1} = G_{s_2}$ for all $s_1, s_2 \in X$ .
(b) There are exactly 8 elements of order 3 in S₄.
(c) If $F = \mathbb{Q}(\sqrt[5]{2}, \sqrt[3]{5})$ , then $[F : \mathbb{Q}] = 8$ .
(d) $\mathbb{F}_7(\sqrt{3}) = \mathbb{F}_7(\sqrt{5})$ .
(e) For any $\alpha\in\mathbb{F}_{25}^{<em>}$ , $\alpha \neq 1, \mathbb{F}_{25} = \mathbb{F}_5[\alpha]$ .