Group Theory Questions - Permutations, Homomorphisms, and Isomorphisms

1. Q.1 Let 6 be an element of the permutation group S5. Then the maximum possible order of 6 is A] 5 B] 6 C] 10 D] 15 Ans B

2. Q.2 The number of group homomorphism's from the cyclic group Z4 to the cyclic Z7 is : A] 7 B] 3 C] 2 D] 1 Ans is d

3. Q.3 which one of the following condition on a group G implies that G is abelian? A] The order of G for some prime P. B] Every proper subgroup of G is cyclic. C] Every subgroup of G is normal in G. D] The function f: G Gdefind by f(x) = for all X belong to G is is a homomorphism.

4. Q.4 Which of the following statement is true? A] Z2 + Z3 is isomorphic to Z6 B] Z3 + Z3 is isomorphic to Z9 C] Z4 + Z6 is isomorphic to Z24 D] none of these Ans is A

5. Q. 5 let S3 be the group of permutations of three distinct symbols. The direct sum S3 + S3 has an element of order A] 4 B] 6 C] 9 D] 18 Ans b

6. Q.6 Let G be a non-abelian group let a belong to G have order 4 and let β belong to G have order 3. Then the order of the element αβ in G. A] is 6 B] is 12 C] is of the form 12k for k>=2 D] need not be finite. Ans is D

7. Q. 7 Let A be a non-empty subset of R Let I(A) denote the set of interior points of A. The I(A) can be A] empty B] singleton C] a finite set contain more than one element D] countable but not finite Ans is A

8. Q.8 The maximum order of a permutation 6 in the symmetric group S10 is……. A] 10 B] 15 C] 25 D] 30 Ans is D

9. Q. 9 The radius of convergence of the power series is ……. A] 0 B] 2 C] 1 D] none of theses Ans is C

10. Q.10 The number of generator of the additive group Z36 is equal to A] 6 B] 12 C] 18 D] 36 Ans is B

11. Q.11 Let G be a cyclic group of order 24. The total number of group isomorphism's of G onto itself is A] 7 B] 8 C] 17 D] 24 Ans is B

12. Q.12 Let G be a group of order 17. The total number of ono-isomorphic subgroup of G is… A] 1 B] 2 C] 3 D] 17 Ans is B

13. ajjjsjsjk Ans is A

14. Q.14 The function of which the power series converges is A] a B] 1 C] 2 D] none of these Ans is B

15. Q.15 Which one of the following is true ? A] The characteristic of the ring 6z is 6 B] The ring 6z has a zero divisor C] The characteristics of the ring () x 6z is 0 D] The ring 6z x 6z is an integrant domain Ans is B

16. Q.16 If the power series n convergence for x=3 then the series n A] convergent absolutely for x = - 2 B] convergent but not absolutely for x = - 1 C] convergent but not absolutely for x = 1 D] divergent for x = - 2 Ans is A

17. Q.17 Which of the following condition not ensure the convergence of real sequence { an } ? A] | an – an + 1 |→ 0 as n→ B] is convergence C] is convergence D] The sequence { a3n } are convergence Ans is A

18. Q.18 The radius of convergent of the power series A] 0 B] C] 3 D] Ans is B

19. Q .19 The number of all subgroup of the group ( Z60 ,+ 60 ) is A] 2 B] 10 C] 12 D] 60 Ans is C

20. Q.20 Which of the following sequence of function is uniformly convergent an ( 0 , 1 ) ? A] B] C] D] none of these Ans is D

21. Q.21 G = { 1, -1, i, -i } with multiplication is an abelian group or not? Abelian Group

22. Q.22 If G is a finite group and a belong to G then = --------- A] a B] 0 C] 1 D] e Ans I D

23. Q.23 for n>=3 the permutation group Sn is --------- A] abelian B] nonablian C] both A and B D] none of these Ans is B

24. Q.24 The product of two even or two odd permutation is……… A] odd B] even C] Both A and B D] none of these Ans is B

25. Q.25 An identity permutation is always _____________ permutation. A] odd B] even C] both A and B D] none of these Ans is B

26. Q.26 Consider the group S9 of aU permutation on a set with 9 element what is the largest order of permutation in S9 _______ A] 10 B] 20 C] 30 D] 40 Ans is B

27. Q.27 Every group of prime order is _______ A] simple B] mixed C] Both A and B D] none of these Ans is A

28. Q.28 Let G be the group of order 17 The total number of non isomorphic subgroup of G is _____ A] 4 B] 3 C] 2 D] 5 Ansis C