BSc Mathematics SEM IV 2019 20 Oct 2019-20 ATKT MATHS II Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q2 Figures to right Indicate full marks
2. Q1 Choose correct alternative in each of the following (2 marks each)
3. Q1 Which of the following set is group under indicated binary operation
4. Q2 Inverse of
   • a) Need not unique b) is unique may be two d) None of these Ina group G, the number of element at G such that
   • b)1 of these
5. Q4 The group of symmetries of a regular n-gon (n>3) has
   • a) of order 2 & n-1 elements of order
   • b)pyelements of order 2 if nis odd
   • c) Exactly 2 elements of order n
   • d) None of these
6. Q5 The number of elements of order 2 in Sy is
   • a)8 b)6 None of these
7. Q6 Let H be a subgroup of G. G if aH+bH then
   • a) b) aHMbH #@ c)aHCbH d) None of these
8. Q7 The left cossets of in U (30) are
   • c) H,1+H,29+H d) None of these
9. Q8 Order of U(n), n>2 is
   • a) Even b) odd d) None of
10. Q9 Number of homomorphisms from Z to
    • a) 6 d) None of these
11. Q10 The group of symmetries of
    • a) a square is abelian b) an equilateral tridngle is abelian is abelian None of these
12. Q2 a) Attempt any ONE question from the following (8 marks each)
13. Q1 Let G be a group then prove that
    • i) Identity elements of G is unique
    • ii) Inverse of an element in G is unique Cancellation laws holds in G iv)
14. Q2 State and prove necessary and sufficient condition for a nonempty set to be a subgroup
15. Q2 b) Attempt any two question from the following (6 marks each) 1)(Z “ny a group is prime, prove it
16. Q2 let H be a finite subset of a group (G,*) then prove that H is a subgroup of G iff
17. Q3 Let G be a group, if o(a)=n then prove that where (m,n)= gcd of
18. Q4 If G is group of even order then show that G has an element of order two
19. Q3 (a) Attempt any one questions from the following (8 marks each)
20. Q1 Let G be a finite cyclic group of order n generated by a then prove that a" is also generator of
21. Q2 Define a cyclic group and prove that every finite cyclic group of order n has unique subgroup of order d for each divisor d of n
22. Q3 (b) Attempt any two questions from the following (6 marks each) Let G be an infinite cyclic group generated by a. show that every nontrivial subgroup of G an infinite cyclic and prove or disprove following is a group under operation (a,b)(c,d)=(ac,bd) but not a cyclic group
23. Q2 Prove that every subgroup of cyclic group is cyclic
24. Q3 Let G be a finite group then prove that a is generator of G iff
25. Q4 Show that every subgroup of prime order P is cyclic. further show that it has (P-1) generators
26. Q4 (a) Attempt any one questions from the following (8 marks each)
27. Q1 State and prove Lagrange’s theorem for finite group
28. Q2 Define kernel of group homomorphisms, prove that if is group homomorphisms then show that kerf is subgroup of G and show that f injective iff ker f={e}
29. Q4 (b) Attempt any two questions from the following (6 marks each)
30. Q1 Define an automorphism of group . let show that fa: defined by VxEG is an automorphism
31. Q2 State and prove fermat’s Little theorem
32. Q3 Let H be a subgroup of group G,a. then prove that
33. Q4 Ina finite group, show that the order of each element of the group decides the order of the
34. Q5 Attempt any four questions from the following. (5 marks each )
35. Q1 List all generators, all subgroups of cyclic group (Zis,+)
36. Q2 Show that the group U(8) is not isomorphic to U(10) but U(8) is isomorphic to U 12 marks
37. Q3 i) Find all subgroups of Klein’s four group
    • ii) Prove that every proper subgroup of s3 is cyclic
38. Q4 Let H be a subgroup of a group.G then prove that
39. Q5 Let H bea subgroup of group G & then prove that HUk is a subgroup of G iff either HCK or
40. Q6 i) Let Hbe a subgroup of group G & then prove that xH=H iff xEH
    • ii) Let be subgroup of group G then prove that HNK is also a subgroup of G

Read from the scan above, so a character or two may differ. The scan is the original.