UCME Algebraic Structures B.Sc Question Paper : ideunom.ac.in

University : University of Madras
Degree : B.Sc
Department :Mathematics
Subject :UCME Algebraic Structures
Document type : Question Paper
Website : ideunom.ac.in

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IDEUNOM UCME Algebraic Structures Model Paper

U/ID 32355/UCME
Time : Three hours
Maximum : 100 marks

Related / Similar Question Paper : IDEUNOM B.Sc PAH Dynamics Question Paper

OCTOBER 2011

PART A — (10 × 3 = 30 marks)
Answer any TEN questions.
Each questions carries 3 marks.
1. Define permutation group.
2. If f is a homomorphism of G onto G . Prove that f (e) = e , where e and e are unit elements of G
3. State Cauchy’s theorem for abelian groups.
4. Define integral domain.
5. Define maximal ideal.
6. If S is a subsets of V . Prove that L (L (S)) = L (S).
7. Define orthogonal vectors.
8. Define orthonormal vectors.
9. Define characteristic root.
10. Define Rank of matrix.
11. Define regular and singular elements in A (V ).
12. Define a linearly dependent set of vectors in a vector space.

PART B — (5 ´ 6 = 30 marks)
Answer any FIVE questions.
Each questions carries 6 marks.
13. If G is a finite group and N is a normal subgroup of G, prove that ( ) ( ) O (N) O G O G /N = .
14. If O (G) = P2 where P is a prime number, prove that G is abelian.
15. If R is a ring, prove that for all a, bÎR (a) 0a = a0 = 0 (b) a (- b) = (- a)b = -(ab) (c) (-1)a = -a .
16. If V is a finite dimensional vector space, prove that it contains a finite set n v , v ….v 1 2 of linearly  independent elements whose linear span is V
17. If u, vÎV then prove that (u, v) £ u v .
18. If V is finite dimensional over F, prove that T Î A (V ) is regular iff T maps V onto V
19. If V is finite dimensional over F prove that (a) r (ST )£ r (T ) (b) r (TS)£ r (T ).

PART C — (4 ´ 10 = 40 marks)
Answer any FOUR questions.
Each questions carries 10 marks.
20. State and prove fundamental theorem of group homomorphism.
21. (a) Let R be a commutative ring with unit element whose only ideals are {0 } and R itself. Prove that R is a field.
(b) If F is a field then prove that it has two ideals {0 } and F itself.
22. If n v , v ….v 1 2 is a basis of V over F and if w , w ….w 1 2 in V are linearly indenpendent over
F, prove that m £ n .
23. State and prove Gram-Schmidt orthogonalization process.
24. If A and B are finite dimensional subspaces of a vector space V, prove that (A + B) is finite dimensional and dim(A + B) = dim A + dimB
25. If V is n-dimensional vector space over F and if prove T satisfies a polynomial of degree n over F.

OCTOBER 2012

U/ID 32355/UCME
Time : Three hours
Maximum : 100 marks
PART A : (10 × 3 = 30 marks)
Answer any TEN questions.
Each question carries 3 marks :
1. Let G be the group of integers under addition and let f :G ®G be defined by f (x)=2x . Prove that f is a homomorphism.
2. Express the permutation (1,2), (1, 2, 3)(1, 2) as a product of disjoint cycles.
3. Prove that every field is an integral domain.
4. If U is an ideal of R and 1ÎU , prove that U = R .
5. Define an Euclidean ring.
6. Define the linear span of a non empty subset S of a vector space V and prove that it is a subspace of V .
7. If V is a vector space over F and if u,v ÎV and a ÎF , prove that a (u – v)=au -av .
8. If F is the field of real numbers, find A (W) where W is spanned by (1, 2, 3) and (0,4, -1)
9. If T is a linear transformation with rank of T ,r (T )= 0 , what can you say about T ?
10. If l ÎF is a characteristic root of T ÎA (V ), prove that there exist v ¹ 0 in V such that vT = lv .

PART B : (5 × 6 = 30 marks)
Answer any FIVE questions.
Each question carries 6 marks :
13. State and prove fundamental theorem of homomorphism.
14. Prove that the alternating group n A is a normal subgroup of n S and [ : ]= 2 n n S A .
15. If R is a commutative ring with unit element and M is an ideal of R, prove that M is a maximal ideal of R if and only if R M is a field.

16. State and prove unique factorisation theorem.
17. If V is a finite-dimensional vector space over F , prove that any two bases of V have the same number of elements.
18. Derive Schwarz inequality.
19. If V is finite-dimensional over F , prove that T ÎA (V ) is invertible if and only if the constant term of the minimal polynomial for T is not 0.

PART C : (4 × 10 = 40 marks)
Answer any FOUR questions.
Each question carries 10 marks :
20. State and prove Cayley’s theorem.
21. Derive the class equation of a group G . Using the class equation, prove that if p is a prime number and p|O(G), then G has an element of order p .
22. If U is an ideal of a ring R, prove that R U is a ring and is a homomorphic image of R.
23. Prove that every integral domain can be imbedded in a field.
24. If V is a finite-dimensional inner product space and if W is a subspace of V , prove that V =W +W^ .
25. If V is n -dimensional over F and if T ÎA(V ) has all its characteristic roots in F , then prove that T satisfies a polynomial of degree n over F