Kakatiya University (KU) 2010-2nd Year B.Sc Mathematics B.A./ (ABSTRACT ALGEBRA & REAL ANALYSIS) - Question Paper

KAKATIYA UNIVERSITY

B.A./B.Sc II YEAR MATHEMATICS

(2009-2010)

paper-II

ABSTRACT ALGEBRA & REAL ANALYSIS

ques. BANK FOR PRACTICAL exam



UNIT-I


1. a) Show that the set S = ( {1, 3, 5, 7}, X8) forms a group.

b) Show that every group G with identity e and such that x * x = e for all

x I G is abelian.

2. obtain the order of the cyclic subgroup of U6 generated by cos 2p/3 + i sin 2p/3.



3. obtain the order of the cyclic subgroup of the multiplicative group G of invertible

four x four matrices generated by


4. calculate the subgroups <1>, <2>, <3>, <4> and <5> of the group Z6.

5. a) obtain the number of generators of a cyclic group having the order 60.

b) obtain the number of automorphisms of the group Z6.


6. obtain the number of elements in the cyclic subgroup of the group C* of nonzero complex numbers generated by one + i.

7. obtain all subgroups of the group Z12, and draw the subgroup diagram for the subgroups of Z12.

8.

9. obtain all orders of subgroups of the group Z12. Also, obtain the elements in every subgroup.


10. If s = , Z =

then obtain i) s -1 Z s ii) | < s > |



11 a) obtain all orbits of

b) Express the subsequent permutation as a product of disjoint cycles, and then as a product of transpositions




12. a) obtain all orbits of the permutation s, where s : Z ® Z is provided by
s (n) = n + one

b) What is the order of Z = (1, 4) (3, 5, 7, 8) ?

13. obtain all cosets of the subgroup 4Z of Z. Also, obtain (Z : 4Z).

14. obtain all cosets of the subgroup < four > of Z12. What is the index of < four > in Z12.


15. obtain the index of < three > in the group Z24.

16. Let s = (1, 2, 5, 4) (2, 3) in S5. obtain the index of < s > in S5.

17. a) Let f : R* ® R* under multiplication be provided by f (x) = | x |. Show that f is a homomorphism.

b) Let ker (f) and f (25) for f : Z ® Z7 such that f(1) = 4.

18. a) Let f : R ® R*, where R is additive group and R* is multiplicative group, be provided by f (x) = 2x. Examine whether f is a homomorphism or not.

Earning:   Approval pending.