Practice Questions on Group and its Basic Properties with Answers

**Questions on Group and its Basic Properties with Answers**

**Q1. Let σ : {1,2,3,4,5} → {1,2,3,4,5} be a permutation (one-to-one and onto function) such that σ ^{-1} (j) ≤ σ^{-1} (j) ∀j, 1 ≤ j ≤ 5. Then which of the following are true?**

(a) σ•σ(j) = j for all j , 1 ≤ j ≤ 5.

(b) σ^{-1} (j) ≤ σ^{-1}** ** for all j, 1 ≤ j ≤ 5

(c) The set {k : σ(k) ≠ k} has an even number of elements.

(d) The set {k: σ(k) = k} has an odd number of elements.

**Q2. Which of the following is Non-cyclic group**

(a) (pZ,+); where p is prime.

(b) (mZ,+); where m is integer.

(c) (R^{*},*); where R* = R – {0}

(d) None of the above.

**Q3. Let C* denote the multiplicative group of non-zero complex numbers. Let G _{1} be the cyclic subgroup generated by 1+i and G_{2} be the cyclic subgroup generated by (1+i)/√2. Which one of the following is correct?**

(a) Both G_{1} and G_{2} are infinite groups

(b) G_{1} is finite, but G_{2} is infinite group

(c) G_{1} is finite, but G_{2} is infinite group

(d) Both G_{1} and G_{2} are finite groups

**Q4. Let σ be an element of the permutation group S _{5}. Then the maximum possible order of σ is**

(a) 5

(b) 6

(c) 10

(d) 15

Q5. Let the group G = R ( the set of all real numbers) under addition and the group H =R^{+} ( the set of all positive real numbers) then under multiplication.

(a) H is a cyclic group and G is a non- cyclic group.

(b) G is a cyclic group and H is a non- cyclic group

(c) Neither G nor H cyclic

(d) Both G and H are cyclic

**Q6. Suppose that H is the smallest subgroup of Z under addition and H contains 18, 80, 40then H is**

(a) 18 Z

(b) 40 Z

(c) 2 Z

(d) Z

**Q7. If the order of every non-identity element in a group is ‘n’, then**

(a) ‘n’ is necessarily a prime number

(b) ‘n’ can be any odd number

(c) ‘n’ is an even number

(d) ‘n’ can be any positive

**Q8. Let G={g _{1},g_{2},…,g_{n}} be a finite group and suppose it is given that g_{i}^{2} = identity, for i = 1,2,…,n-1. Then**

(a) g_{n}^{2} is identity and G is abelian

(b) g_{n}^{2} is identity, but G could be non- abelian

(c) g_{n}^{2} may not be identity

(d) None of the above

**Q9. Consider the multiplicative group G of all the (complex)2 ^{n} -th roots of unity where n = 0,1,2.. Then**

(a) Every proper subgroup of G is finite.

(b) G has a finite set of generators

(c) G is cyclic

(d) Every finite subgroup of G is cyclic

**Q10. Which of the following numbers can be orders of permutations σ of 11 symbols such that σ does not fix any symbol?**

(a) 18

(b) 30

(c) 15

(d) 28

**Q11. Let G =Z _{10} x Z_{15}. Then**

(a) G contains exactly one element of order 2

(b) G contains exactly 5 elements of order 3

(c) G contains exactly 24 elements of order 5

(d) G contains exactly 24 elements of order

**Q12. Which of the following groups has a proper subgroup that is NOT cyclic?**

(a) Z_{15} x Z_{17}

(b) S_{3}

(c) (Z,+)

(d) (Q.+)

**Q13. Let G _{1} be an abelian group of order 6 and G_{2} = S_{3} . For j = 1,2 . let P be the statement:**

**“G _{j} has a unique subgroup of order 2″**

**Then**

(a) both P_{1} and P_{2} hold

(b) neither P_{1} nor P_{2} hold

(c) P_{1} holds but not P_{2}

(d) P_{2} holds but not P_{1}

**Q14. Let D _{8} denote the group of symmetries of square (dihedral group). The minimal number of generators for D_{8} is**

(a) 1

(b) 2

(c) 4

(d) 8

**Q15. Consider the following statements.**

**S: Every non abelian group has a nontrivial abelian subgroupT: Every nontrivial abelian group has a cyclic subgroup. **

**Then**

(a) Both S and T are false

(b) S is true and T is false

(c) T is true and S is false

(d) Both S and T are true

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*Answer Key*

*Answer Key*

01. | (a), (b), (c), (d) | 06. | (c) | 11. | (a), (c), (d) |

02. | (c) | 07. | (a) | 12. | (d) |

03. | (c) | 08. | (a) | 13. | (c) |

04. | (b) | 09. | (a), (d) | 14. | (b) |

05. | (c) | 10. | (a), (b), (c), (d) | 15. | (d) |