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IGNOU MMT 3 Algebra Solved Assignment 2024

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Rs. 50

Last Date of Submission of IGNOU MMT-03 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Semester Wise
January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam).
July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).

Title Name	IGNOU MMT 3 Algebra Solved Assignment 2024
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	2024
Session	-
Language	English Medium
Assignment Code	MMT-03/Assignmentt-1//2024
Product Description	Assignment of MSCMACS (M.Sc. Mathematics with Applications in Computer Science) 2024. Latest MMT 03 2024 Solved Assignment Solutions
Last Date of IGNOU Assignment Submission	Last Date of Submission of IGNOU MMT-03 (MSCMACS) 2024 Assignment is for January 2024 Session: 30th September, 2024 (for December 2024 Term End Exam). Semester Wise January 2024 Session: 30th March, 2024 (for June 2024 Term End Exam). July 2024 Session: 30th September, 2024 (for December 2024 Term End Exam).
Assignment Code	MMT 3/2024

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Rs. 50

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 ?, then $G_{s_1} = G_{s_2}$ for all $s_1 , s_2 \, \epsilon \, X$ .

(b) There are exactly 8 elements of order 3 in $s_4$ .

(d) $F_7 (\sqrt{3}) = F_7 (\sqrt{5})$ .

(e) For any $\alpha \, \epsilon \, F_{2^5}^*, \alpha = 1 , F_{5^2}^* = F_2[\alpha]$

Ques 2.

Consider the natural action of $GL_2(R)$ on $M_2(R)$ , the set of $2\times 2$ real matrices, by left multiplication.

(i) Under this action, if $det(x) \neq 0$ , show that the stabiliser of $x \in M_2 (R)$ is {I}, where I is the 2 × 2 identity matrix.

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

(iii) 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 $b = \begin{bmatrix} u && v\\ w && z \end{bmatrix}$ such that $b = \begin{bmatrix} a'\\ b' \end{bmatrix} = 0$ and $b \begin{bmatrix} a'\\ b' \end{bmatrix} = \alpha \begin{bmatrix} a'\\ b' \end{bmatrix}$ .(Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)

(iv) Check that I−b is in the stabiliser of x. Also, show that there are infinitely many choices of $\alpha$ for which I − b is invertible.

Ques 3.

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 4.

Find the elementary divisors and invariant factors of $Z_8 \times Z_{12} \times Z_{15}$ .

Ques 5.

Describe the set of primes p for which $x^2-11$ splits into linear factors over $Z_p$

Ques 6.

Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes

Ques 7.

Is there a finite group with class equation 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2?

Ques 8.

Compute the following:

a). $\bigl( \frac{173}{211}\bigr)$

b). $\bigl( \frac{167}{239}\bigr)$

Ques 9.

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

Ques 10.

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

Ques 11.

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

Ques 12.

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

Ques 13.

By looking at the factorisation of $x^9 - x \in F_3[x]$ guess the number of irreducible polynomials of degree 2 over $F_3$ . Find all the irreducible polynomials of degree 2 over $F_3$ .

Ques 14.

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

Ques 15.

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:

$A^tD - C^tB = I$

$A^tC - C^tA = 0$

$B^tD - D^tB = 0$

(Hint: Use block matrix multiplication.) Also, check that the matrix $B^tD - D^tB = 0\begin{bmatrix} 0 & -A\\ A& 0 \end{bmatrix}$ , where A is a $n\times n$ orthogonal matrix, is a symplectic matrix.

Ques 16.

The aim of this exercise is to show that SP₂(R)acts transitively on R² \ {0}.

Ques 17.

Show that a matrix $\begin{bmatrix} a &b \\ c& d \end{bmatrix} \in GL_2(R)$ is symplectic if and only if ad - bc = 1

Ques 18.

Show that, to prove that SP₂(R) acts transitively on GL₂(R), it is enough to show that, for any vector $\begin{bmatrix} a\\ b \end{bmatrix} \neq 0 \in R^2$ , 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 $A\begin{bmatrix} I \\ 0 \end{bmatrix}$ ?)

Ques 19.

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 20.

In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group $D_n = {x,y :x^n,y^2,yxyx}, n \in N, n\geq 2$

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

(b) Prove the relation $y^ix^jy^kx^l = \left\{\begin{matrix} y^ix^{j+1} & if\: k\: is \: \: even \\ y^{i+k}x^{l-j} & if\: k\: is \: \: odd \end{matrix}\right.$

Further, find all the elements of order 2 in D_n .

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