IGNOU MMT 3 SOLVED ASSIGNMENT

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MMT 3 2025 - English

Course Code: MMT-003

Assignment Code:MMT-003/TMA/2025

Maximum Marks: 100

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_s1 = G_s2 for all s₁, $₂ ∈ X.

(b) There are exactly 8 elements of order 3 in S₄.

(c)

(d)

(e) For any

2. (a) Consider the natural action of GL₂ on M₂, the set of 2 x 2 real matrices, by left multiplication.

(i) Under this action, if det(x) ≠ 0, show that the stabiliser of x ∈ M₂ is {I}, where I is the 2 x 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 GL₂. So suppose x ≠ 0. Let us writeThen,  for non-zero λ ∈ R. Why?

(iii) Let  be a vector that is not a scalar multiple of  . Show that there is a matrix b =  such that b  = 0 and b  = α  (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 a for which I - b is invertible.

(b) 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 Pis normal in K.

(c) Find the elementary divisors and invariant factors of 

3. Describe the set of primes p for which x² - 11 splits into linear factors over Zp. 

4. (a) 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:

(i)             (ii)      

5. (a) Let ? (?) be a finite extension F of odd degree(greater than 1). Show that ? (?²) = ? (?).

(b) Let ? ⊂ ? and let ?, ? ∈ ? be algebraic over F of degree m and n, respectively. Show that [? (?, ?) ∶ ? ] ≤ ??. What can you say about [? (?, ?) ∶ ? ] if m and n are coprime?

(c) Find where ?³ = 1, ? ≠ 1.

6. (a) If char(F) ≠ 2, show that a polynomial ax² + bx + c is irreducible iff   where is the group of squares in .

(b) By looking at the factorisation of x⁹ - x ∈  [x] guess the number of irreducible polynomials of degree 2 over . Find all the irreducible polynomials of degree 2 over .

(c) If Fis a finite field show that there is always an irreducible polynomial of the form x³ - x + a where a ∈ F.(Hint: Show that is not a surjective map.)

7. (a) Suppose that is 2n × 2n matrix where A, B, C and D are nxn matrices. Show that M is symplectic if and only if the following conditions are satisfied:

A^tD - C^tB 1

A^tC - C^tA = 0

B^tD-D^tB = 0

(Hint: Use block matrix multiplication.)

Also, check that the matrix  where A is a n x n orthogonal matrix, is a symplectic matrix

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

(c) Show that

(i) Show that a matrix is symplectic if and only if ad - bc = 1.

(ii) Show that, to prove that SP₂ acts transitively on GL₂, it is enough to show that, for any vector , there is a 2 x 2 symplectic matrix with  as the first column . (Hint: For any matrix A, what is  ?)

(iii) Complete the proof by showing that, given any non-zero  vector non-zero vector, there is always a  such that  is symplectic.

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

(a) Let p be an odd prime that divides n, n = p^r l, p + 1. Suppose C = (x¹). Show that C is the unique Sylow p-subgroup of D_n.

(b) Prove the relation

               

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

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

(d) Suppose n is even, n = 2^km, where 2 + m, k ≥ 2. Let N = (x^m) and H = (y). Show that H N is a subgroup of D_n. What is its order?

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

(a) Let . Show that G is the cyclic group of order six.

(b) Solve the following set of congruences:

(c) Show that is not a UFD by giving two different factorisations of 20.

Question no.	Block 1	Block 2	Block 3	Block 4	Block 5
2 a)	5
2 b)	3
2 c)		2
3 c)			10
4 a)	4
4 b)	3
4 c)			3
5 a)				2
5 b)				5
5 c)				3
6 a)				2
6 b)				6
6 c)				2
7 a)		4
7 c)		1
7 c)		3
7 c)		2
8 a)	3
8 b)	4
8 c)	2
8 d)	3
8 e)	3
9 a)		5
9 b)			5
9 c)			5
Total	30	17	23	20	0

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