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Posted By: Kishori Mohon Dutta Member Level: Silver Posted Date: 28 Jun 2008
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2007 Indira Gandhi National Open University (IGNOU) BACHELOR'S DEGREE PROGRAMME Question paper
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BACHELOR'S DEGREE PROGRAMME
Term-End Examination
ELECTIVE COURSE:MATHEMATICS
MTE-6:ABSTRACT ALGEBRA
Time: 2 hours
Maximum Marks: 50
Note: Attempt five questions in all question no.7 is compulsory. Answer any four question from questions no.1 to 6. No calculators are allowed
Important Note: Some text in this page is not rendered in the correct manner. Please try later.
December, 2007
1.(a) Find all subgroups of Z of which 30Z is a subgroup (3)
(b) Let C[o,1] be the ring of all real valued continous functiions on [0,1],and let M={f?C[0,1]|f(0)=0}.
(i) Show that M is an ideal of C[0,1].
(ii) Is the map f:C[0,1]->R given by f(f)=f(0) a ring homomorphism? Give reasons for your answer.
(iii) Check whether M is a maximal ideal of C[0,1]. (7)
2.(a) Let R be a commutative ring with unity, and I and J be the ideals of R such the I+J=R. Show that ?x?R such that X-a ? I and X-b ? J; Where a, b are given real numbers.5
(b) Let G be a finite cyclic group. If a positive integer m is a divisor of O(G), is it true that G has a subgroup of order d? Justify your answer.3
(c) If f=(1 2 4)(3 5)(6 8 7 9) is a permutation in S10, Write f-1 as a product of disjoing cycles.2
3.(a) Let G be a group.To each g?G, define the map f:G->G by fg(x)=gxg-1 for all X?G. Show that fg is an automorphism of G, and the map f:G->Aut G, given by f(g)=fg,is a group homomorphism. What is kerf?
(b) Let R be a ring and f(x) be a non-zero polynomial in R[x]. Can f(xX) have more roots in R than its degree? Give reasons for your answer. (2)
(c) Give an example of a finite group G with 3 non-trivial proper subgroups H1,H2,H3 such that Hi €Hj for i?j and G=H1 U H2 U H3. 3
4.(a) If G is finite commutative group of order n and if a prime p divides n, show that the number of sylow-p subgroups of G is one. Fine the unique sylow-2 and sylow-3 subgroups of the cyclic group Z24.4
(b) Which of the following are field extensions of Q? Give resons for your answers.
(i) QXQ(ii)Q[X]/<2x3-6x+3>(iii)C[x]/ Further, those that are field extensions of Q, what is their characteristic?6
5.(a) Show that 2-v-5 is an irreducible element of Z[v-5]={a+bv-5/a,b ?Z} but is not a prime element of Z[v-5].4
(b) If G is a group of order 360 and A and B are sylow-2and sylow-3 subgroups of G, find the order of the subgroup AnB. 3
(c) Consider a function f:A->{1}subsetZ.What are the conditions on A if 3
(i) f is to be 1-1?(ii)f is to be surjective?3
6.(a) Let I be an ideal of a ring R. Use the Fundamental Theorem of Homommorphism to prove that M3(R/I)~M3(R)/M3(I)6
(b) On the set of integers Z,define Xfy=X+Y-7. Check whether Z is a group with respect to f, or not. 4
7. Which of the following statements are true? Give reasons for your answers.10
(i) If p and q are distinct prime numbers and G is a group of order pq, then G is cyclic.
(ii) If X={1,2,3,...,10} and a relation ~ is defined on X as a~b if and only if either a divides b or b divides a, then the relation ~ is an equivalence relation on X.
(iii) Every element of sn has order at most n.
(iv) There is no non-trivial ring homomorphism from Z2 to Z3.
(v) There is an integral domain of order 6
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