Kerala University 2005 B.Sc Mathematics (MATHS) - Question Paper
B.SC (MATHS)
Beg. No----------------------------------------------(Pages : 2) K 5103
Name_____................._____.___________
FINAL YEAR B.Sc. DEGREE EXAMINATION, MARH/APRIL 2005
Part IIIGroup IMathematics
Paper. IllALGEBRA . ' .
Time : Three Hours Maximum : 65 Marks
Maximum marks for each unit is 13.
Unit I
1. Define a relation (R on z by setting n(fen if and only if nm > 0. Verify whether (R is an equivalence relation on z.
(3 marks)
ab
2. Define * on Q+ by o. * b = . Prove that (Q+ *) is a group. (4 marks)
3. Let a e G, G a group. Prove that H = jo : n'e zj is a subgroup of G and that it is the smallest subgroup of G that contains a.
_______ ___________________ _ ____j-__ ...-,----- -------(4 marks) -
4. Prove that every group is isomorphic to a group of permutations. (5 marks)
(1 2 3 4 5 6 7 8V
5. Express 3 14 7 2 5 8 6 4 product of transpositions. (4 marks)
Unit II
6. Show that every group of prime order is cyclic. (5 marks)
7. Compute the factor group z4 x z6/{(0,1)). Verify whether it is isomorphic to z4.
(5 marks)
HN
8. If H is a subgroup of G and N is a normal subgroup of G show that H/(H n N).
(5 marks)
9. Let X be a G-set. Define the isotropy subgroup Gx of x e X. Prove that Gx is a subgroup of G
for each x e X.
(5 marks)
Unit HI
10. Prove that zn under addition modulo n and multiplication modulo n form a ring. Verify whether it is an integral domain.
(5 marks)
11. Define the characteristic of a ring R. Prove that if R is a ring with unity then R has characteristic n > 0 if and only if n is the smallest positive integer such that n 1 = 0.
(5 marks)
12. Show that the quartercrions form a skew field under addition and multiplication.
(5 marks)
13. Prove that any two fields of quotients of an integral domain are isomorphic. (5 marks)
Unit IV
14. Let A, B be ideals of a ring R. Define
A + B = {a + 6/a e A, 6 e B).
Show that A + B is an ideal of R.
(5 marks)
R
15. Define a prime ideal of a ring. Prove that an ideal N * R is prime if and only if is an integral domain where R is a commutative ring with unity.
(5 marks)
16. Prove that a non-zero polynomial f (*) e F [ac] of degree n can have at most n zeroes in a field F.
(5 marks)
17. If F is a field prove that every non-constant polynomial fix) e F [x] can be expressed uniquely as a product of irreducible polynomials.
(5 marks)
UnitV
18. Let V be a vector space over F and F1 be a subfield of F. Show that V is a vector space over F1 also.
(5 marks)
19. Show that if Vx is a subspace of V2 and V2 is a subspace of V, then Vx is a subspace of V.
(5 marks)
20. Prove that
dim (Vj + V2) = dim Vx + dim V2 - dim (Vx n V2),
. where Vx and V2 are subspaces of a vector space V.
(5 inarks)
21. Prove that if V, V1 are vector spaces over a field F then the set of all linear transformations of V to V1 form a vector space over F.
(5 marks)
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