IGNOU MMT 3 SOLVED ASSIGNMENT

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IGNOU MMT 3 SOLVED ASSIGNMENT

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Rs. 123

IGNOU MMT 3 SOLVED ASSIGNMENT

~~Rs. 200~~
Rs. 123

Last Date of Submission of IGNOU MMT-03 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).
Semester Wise
January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam).
July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

Title Name	IGNOU MMT 3 SOLVED ASSIGNMENT
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	MASTER DEGREE PROGRAMMES
Course Code	MSCMACS
Course Name	M.Sc. Mathematics with Applications in Computer Science
Subject Code	MMT 3
Subject Name	Algebra
Year	2026
Session
Language	English Medium
Assignment Code	MMT-03/Assignmentt-1//2026
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2026. Latest MMT 03 2026 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-03 (MSCMACS) 2026 Assignment is for January 2026 Session: 30th September, 2026 (for December 2026 Term End Exam). Semester Wise January 2026 Session: 30th March, 2026 (for June 2026 Term End Exam). July 2026 Session: 30th September, 2026 (for December 2026 Term End Exam).

~~Rs. 200~~
Rs. 123

Questions Included in this Help Book

Ques 1.

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 $\alpha\in\mathbb{F}_{25}^{}$ , $\alpha \neq 1, \mathbb{F}_{25} = \mathbb{F}_5[\alpha]$ .

Ques 2.

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.

Ques 3.

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 ?

Ques 4.

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.)

Ques 5.

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.

Ques 6.

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.

Ques 7.

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

Ques 8.

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

Ques 9.

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) $\left(\frac{167}{239}\right)$

Ques 10.

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

Ques 11.

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?

Ques 12.

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

Ques 13.

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

Ques 14.

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 $\mathbb{F}_3$ .

Ques 15.

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.)

Ques 16.

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.

Ques 17.

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

Ques 18.

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.

Ques 19.

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.$

Ques 20.

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.

Ques 21.

) Prove the relation

$$y^i x^j y^k x^l = \begin{cases} y^i x^{j+l} & \text{if } k \text{ is even} \\ y^{i+k} x^{l-j} & \text{if } k \text{ is odd.} \end{cases}$
Further, find all the elements of order 2 in D_n.

Ques 22.

Find all the Sylow 2-subgroups of D_n when n is odd. Describe them in terms of x and y.

Ques 23.

Suppose n is even, $n = 2^k m$ , where $2 \nmid m, k \ge 2$ . Let $N = \langle x^m \rangle$ and $H = \langle y \rangle$ . Show that HN is a subgroup of D_n. What is its order?

Ques 24.

Suppose n is as in the previous part. Find all the Sylow 2-subgroups of D_n. Describe them in terms of x and y.

Ques 25.

Let $G = \langle a, b \mid a^2, b^3, aba^{-1}b^{-1} \rangle$ . Show that G is the cyclic group of order six.
(b) Solve the following set of congruences:
$$\begin{aligned} x &\equiv 2 \pmod{17} \\ 3x &\equiv 4 \pmod{19} \\ x &\equiv 7 \pmod{23} \end{aligned}$
(c) Show that $\mathbb{Q}(\sqrt{-19})$ is not a UFD by giving two different factorisations of 20.

Ques 26.

Ques 27.

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.

Ques 28.

Ques 29.

Ques 30.

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.

Ques 31.

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.

Ques 32.

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

Ques 33.

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

Ques 34.

Ques 35.

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

Ques 36.

Ques 37.

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

Ques 38.

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

Ques 39.

Ques 40.

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.)

Ques 41.

Ques 42.

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

Ques 43.

Ques 44.

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.$

Ques 45.

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.

Ques 46.

) Prove the relation

$$y^i x^j y^k x^l = \begin{cases} y^i x^{j+l} & \text{if } k \text{ is even} \\ y^{i+k} x^{l-j} & \text{if } k \text{ is odd.} \end{cases}$
Further, find all the elements of order 2 in D_n.

Ques 47.

Find all the Sylow 2-subgroups of D_n when n is odd. Describe them in terms of x and y.

Ques 48.

Suppose n is even, $n = 2^k m$ , where $2 \nmid m, k \ge 2$ . Let $N = \langle x^m \rangle$ and $H = \langle y \rangle$ . Show that HN is a subgroup of D_n. What is its order?

Ques 49.

Suppose n is as in the previous part. Find all the Sylow 2-subgroups of D_n. Describe them in terms of x and y.

Ques 50.

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Course Name	M.Sc. Mathematics with Applications in Computer Science
Course Code	MSCMACS
Programm	MASTER DEGREE PROGRAMMES Courses
Language	English

IGNOU MMT 3 Solved Assignment

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