Algebra-I M.Sc Model Question Papers : alagappauniversity.ac.in

Name of the University : Alagappa University
Degree : M.Sc
Department : Mathematics
Subject Code/Name : Algebra-I
Semester : I
Document Type : Model Question Papers
Website : alagappauniversity.ac.in

Download Model/Sample Question Paper : Nov 2010 : https://www.pdfquestion.in/uploads/alagappauniversity.ac.in/3996.-M.SC.MATHS%20CBCS.pdf

Algebra-I :

Time : 3 Hours
Maximum : 75 Marks
Part – A :

Related : Alagappa University Differential Calculus and Trigonometry B.Sc Model Question Papers : www.pdfquestion.in/3931.html

1. Suppose H and K are subgroups of a group G of order 10 and 21 respectively. Find o(HK).
2. Show that the intersection of two normal subgroups of a group G is also a normal subgroup of G.
3. Define the internal direct product of subgroups of a group and express as internal direct product of its proper subgroups.
4. Find the class equation of S3.
5. Show that if a, b in R, then
(i) a0 = 0a = 0.
(ii) a(– b) = – (a)b = – (ab) and
(iii) (– a)(– b) = ab.
6. Prove that any field is an integral domain.
7. If I is a ideal of a ring R containing the unit element, show that I = R.
8. Show that the intersection of two ideals is again an ideal.
9. Prove that an Euclidean ring possess an unit element.
10. Let R be an integral domain with unit element. Prove that if for a, b in R, a|b and b|a then both a and b are associates in R.

Part – B : (5 × 5 = 25)
Answer all questions.
11. (a) Prove that An is a normal subgroup of Sn with index 2.
(Or)
(b) If G is a finite group and H != G is a subgroup of G such that o(G) does not divide i(H), then sh ow t h at H contains a nontrivial normal subgroup of G.
12. (a) Let o(G) = pn, where p is a prime number. Prove that G has a nontrivial center.
(Or)
(b) Show that any two p-sy low subgroups of a group G are conjugate.
13. (a) Show that a finite integral domain is a field.
(Or)
(b) Prove that a ring homomorphism R->R is one to one if and only if the kernel of 0 is zero submodule.
14. Show that the set of multiples of a fixed prime number p form a maximal ideal of the ring of integers.
15. (a) If p is a prime number of the form 4n +1, then prove that p = a2 +b2 for some integers a,b.
(Or)
(b) State and prove the division algorithm for polynomials.

Part – C : (3 × 10 = 30)
Answer any three questions.
16. State and prove the Cauchy’s theorem for abelian groups.
17. Let G be a group and suppose that G is a internal direct product of N1,N2 …Nn. Let T = N1 × … × Nn. Then prove that G and T are isomorphic.
18. Show that the ring of quaternion is a non-commutative division ring.
19. Show that a commutative ring with unit element R is a field if and only if the only ideals of R are (0) and R itself.
20. Show that if R is a unique factorization domain then R[x] is also an unique factorization domain.

M.Sc. Degree Examination, November 2010 :
Analysis—I :
(CBCS—2008 onwards)
Time : 3 Hours Maximum : 75 Marks
Part – A (10 × 2 = 20)
Answer all questions.
1. Define limit point of a set and give an example of a set which has no limit point.
2. Give an example to show that disjoint sets need not be separated.
3. When do you say that a series converges ? Give an example of a divergent series.
4. Define upper and lower limits of a sequences.
5. Give an example of a power series with radius of convergence zero.
6. State the partial summation formula.
7. Give an example of a function which has second kind discontinuity at every point.
8. Monotomic function have no discontinuities of the second kind. Why ?
9. Define the derivation of a real function.
10. Define local maximum and local minimum of a real function defined on a metric space.

Part – B : (5 × 5 = 25)
Answer all questions.
11 (a) Let A be the set of all sequences whose elements are the digits 0 and 1. Prove that A is uncountable.
(Or)
(b) Prove that compact subset of metric space are closed.
12. (a) Prove that if {pn} is a sequence in a compact metric space X, then some subsequence of {pn} converges to a point of X.

Part – C : (3 × 10 = 30)
Answer any three questions.
16. If a set E is Rk has one of the following three properties, then prove that it has the other two :
(a) E is closed and bounded
(b) E is compact
(c) Every infinite subset of E has a limit point in E.
17. State and prove (a) Root test ; (b) Ratio test.
18. Let f be a continuous mapping of a compact metric space X into a metric space Y. Prove that f is uniformly continuous on X.
19. State and prove the L’Hospital’s rule.