Assignment 1 Basic Notions

Assignment 1 Basic Notions

1. The number of positive divisiors of 50,000 is

(a) 20 (b) 30 (c) 40 (d) 50

2. The number of elements in the set {m ∶ 1 ≤ m ≤ 1000, m and 1000 are relatively prime} is

(a) 1000 (b) 250 (c) 300 (d) 400

3. Let U(n) be the set of all positive integers less than n and relatively prime to n for n = 248, the number

of elements in U(n) is

(a) 60 (b) 120 (c) 180 (d) 240

4. Two finite sets m and n elements. The total number of subsets of the first set is 56 more than the total

number of subsets of the second set. Then the values of m and n are respectively.

(a) 8, 5 (b) 6, 3 (c) 4, 1 (d) none of these

5. Let A be a set contains n elements, then its power set P(A) contains

(a) n elements (b) 2n elements (c) n2 elements (d) none of these

6. 20 teachers of a school either teach mathematics or physics. 12 of them teach mathematics while 4 teach

both the subjects. Then the number of teachers teaching physics only is

(a) 12 (b) 8 (c) 16 (d) none of these

7. Let A = {1, 2, 3, 4} and let P be a relation on A given by ℛ is

(a) reflexive (b) symmetric (c) transitive (d) none of these

8. A survey shows that 63 % of Americans like cheese whereas 76% like apples. If x % of the Americans

like both cheese and apples, then

(a) x = 39 (b) x = 63 (c) 39 ≤ x ≤ 63 (d) none of these

9. A set contains 2n + 1 elements. The number of subsets of this set containing more than n elements is

equal to

(a) 2n−1 (b) 2n (c) 2n+1 (d) 22n

10. If two sets A and B are having 99 elements in common, then the number of elements common to each of

the set A × B and B × A are

(a) 299 (b) 992 (c) 100 (d) 18

11. The relation less than in the set of natural numbers is

(a) only symmetric (b) only transitive

(c) only reflexive (d) equivalence relation

12. Let A = {a, b, c} and B = {c, d, e} then the number of both way relation is

(a) 4 (b) 8 (c) 2 (d) 16

13. Let A = {1, 2, , 3, 4}, B = {2, 4, 5} and C = A ∩ B then number of symmetric relation on C is

(a)3 (b) 8 (c) 2 (d) 4

14. If set A is empty then cardinality of the set P (P(P(A))) is

(a) 6 (b) 16 (c) 2 (d) 4

15. If A is the set of evern natural number less than 8 and B is the set of prime numbers less than 7. Then

number of relation from A to B is

(a) 29 (b) 92 (c) 32 (d) 23 − 1

16. If A = {1, 2, 3} then the number of equivalence relation on A is

(a) 3 (b) 7 (c) 5 (d) 8

17. A = {(x, y) ∶ y = ex, x ∈ ℝ}

B = {(x, y) ∶ y = e−x; x ∈ ℝ}

(a) A ∩ B = ϕ (b) A ∩ B ≠ ϕ (c) A ∪ B = ℝ (d) A ∪ B = A

18. If A = {(x, y) ∶ x2 + y2 = 32} and B = {(x, y) ∶ x2 + 3y2 = 12} then A ∩ B contains

(a) two points (b) three points (c) one points (d) four points

19. Let S be a finite set with n elements then the number of symmetric relations is

(a) 2n(n−1) (b) 2n(n+1) (c) 2n(n−1)

2 (d) 2n(n+1)

2

20. In a group of 265 persons, 200 like singing, 110 like dancing and 55 like painting, if 60 person like both

singing and dancing, 30 like both singing and painting and 10 like all three activities then the number of

person who like only dancing and painting is

(a) 10 (b) 20 (c) 30 (d) 40

Multi-Select Questions

1. Consider the following relations in ℤ, then which of the following is an equivalence relation?

(a) m~n if and only if mn ≥ 0 (b) m~n if and only if mn > 0

(c) m~n if and only if |m| = |n| (d) m~n if and only if mn ≤ 0

2. Which of the following is not an equivalence relation in ℝ

(a) x ≤ y ∀ x, y ∈ ℝ (b) x − y is an irrational number

(c) x − y is divisible by 3 (d) x − y is a perfect square

3. Let n be a fixed positive integer. Define a relation ℛ on the set ℤ of integers by aℛb ⇔ n|(a − b) then

ℛ is

(a) reflexive (b) symmetric (c) transitive (d) equivalence

4. Let ℛ be a relation over the set ℕ × ℕ and it is defined by (a, b)ℛ (c, d) ⇔ a + d = b + c then ℛ is

(a) reflexive only (b) symmetric only

(c) transitive only (d) an equivalence relation

Let ℛ be a reflexive relation on a set with n elements and ℛ has m ordered pair elements then;

(a) m > n (b) n ≥ m (c) m = n (d) none of these

5. Let ℕ be the set of natural numbers, define a relation on ℕ as aℛB ⇔ a + b is even ∀ a, b ∈ ℕ then

(a) ℛ is an equivalence relation

(b) these are only two class of with respect to this relation

(c) Cl(2) ∩ Cl(1) = ϕ (d) Cl(3) ∩ Cl(5) = ϕ

6. An integer m is said to be related to another integer n if m is a multiple of n then relation is

(a) reflexive and symmetric (b) reflexive and transitive

(c) symmetric and transitive (d) equivalence relation

7. Which of the following statements is correct?

(a) Empty relation is always reflexive (b) Empty relation is always irreflexive

(c) Empty relation may be reflexive (d) All of the above

8. Let A = {a, b, c} and S = {(a, a), (b, b), (b, c)}. S ⊆ A × A, then S is

(a) reflexive relation (b) symmetric relation

(c) transitive relation (d) none

9. The relation ℛ defined on the set A = {1, 2, 3, 4, 5} by P = {(x, y) ∶ |x2 − y2| < 16} is given by

(a) {(1, 1), (2, 1), (3, 1), (4,1), (2,3)} (b) {(2,2), (3,2), (4,2), (2,4)}

(c) {(3,3), (4,3), (5,4), (3,4)} (d) none of these

10. Which of the following is not an equivalence relation in ℤ?

(a) aℛb ⇔ a + b is an even integer (b) aℛb ⇔ a − b is an even integer

(c) aℛb ⇔ a < b (d) aℛb ⇔ a = b

Previous Year Questions

Net June 2015

1. Let D be the set of touples (w1, w2, ⋯ , w10), where wi ∈ {1, 2, 3}, 1 ≤ i ≤ 10 and wi + wi+1 is an even

number for each i with 1 ≤ i ≤ 9 then the number of elements in D is

2. The number of surjective maps from a set of 4 elements to a set of 3 elements is

(a) 36 (b) 64 (c) 69 (d) 81

Net June 2014

3. An icecream shop sells icecreams in five different flavours. , Vanilla, Chocolate, Strawberry, Mango and

Pineapple. How many combinations of these scoop cones are possible?

[Note: The repition of flavours is allowed but the order in which the flavours are chosen does not matter]

(a) 10 (b) 20 (c) 35 (d)243

Net June 2013

4. Consider the two sets A = {1, 2, 3} and B = {1, 2, 3, 4,5}. Choose the correct statements.

(a) The total number of functions from A to B is 125

(b) The total number of functions from A to B is 243

(c) The total number of one one functions from A to B is 60

(d) The total number of one one functios from A to B is 120.

Net Dec 2012

5. In a group of 265 persons, 200 like singing, 110 like dancing and 55 like painting. If 60 person like both

singing and dancing, 30 like both singing and painting and 10 like all three activities, then the number

of persons who like only dancing and painting is

(a) 10 (b) 20 (c) 30 (d) 40

TIFR 2010

6. The total number of subset of a set of 6 elements is


Answers

Single Correct

1. (b) 2. (d) 3. (b) 4. (b) 5. (b)

6. (b) 7. (c) 8. (c) 9. (d) 10. (b)

11. (b) 12. (c) 13. (b) 14. (d) 15. (a)

16. (a) 17. (b) 18. (d) 19. (d) 20. (a)

Multiple Select Answers

1. (c) 2. (a, b, d) 3. (a, b, c, d) 4. (d) 5. (a, c)

6. (b) 7. (b, c) 8. (d) 9. (d) 10. (c)

Previous Year

1. (b) 2. (b) 3. (c) 4. (a, c) 5. (a)

6. (d)

Group Theory Assignment 3

(Group and Subgroup)

1. 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 these

2. 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(J) ∀ J 1 ≤ J ≤ 5. Then which of the following is are⁄ true?

(a) σ ∘ σ(J) = J for all J, 1 ≤ J ≤ 5 (b) σ−1(J) = σ(J) 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.

3. The order of a and x in a group are respectively 3 and 4. Then the order of x−1ax is

(a) 3 (b) 4 (c) 6 (d) 12

4. If the order of every non identity element in a group is n then

(a) ʻnʼ is necesarily a prime number (b) ʻnʼ can be any odd number

(c) ʻnʼ is an even number (d) ʻnʼ can be any positive

5. Consider the multiplicative group G of all the (complex) 2nth roots of unity where n = 0, 1, 2, ⋯

(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

6. Let G be a finite abelian group and a, b ∈ G with order (a) = m, order (b) = n which of the following

are necessarily true?

(a) O(ab) = mn (b) O(ab) = LCM(m, n)

(c) there is an element of G whose order is LCM(m, n)

(d) O(ab) = gcd (m, n)

7. The number of elements of S5 (The symmetric group on 5 letter) Which are their own inverses equals.

(a) 10 (b) 11 (c) 25 (d) 26

8. Let G be a non abelian group and α, β ∈ G have order 3, 3 respectively. Then the order of the element

αβ ∈ G

(a) 16 (b) 12

(c) is of the form 12k for k ≥ 2 (d) need not be finite

9. Let S10 be a group of permutation of 10 symbols such that β ∈ S10 be 10 cycle and

α = βk , where 2 ≤ k ≤ 10 , then the value of k for which α and β have same order

(a) 3 (b) 4 (c) 5 (d) 6

10. Choose the correct statements

(a) A non cyclic group can have all of its proper subgroup of cyclic.

(b) Every finite cyclic group have even number of generations

(c) infinite cyclic group has precisely two generators.

(d) Every finite group of composite order possesses proper subgroup

11. The order of the group generated by the matrices (0 1

−1 0) and (

0 ii 0

) where i = √−1 under

multiplication is ⋯

12. Let Sn be the group of all permutations on the set {1, 2, ⋯ , n} under the composition of mapping for

n > 2. If H is the smallest subgroup of Sn containing the transposition (1, 2) and the cycle (1, 2, ⋯ , n)

then

(a) H = Sn (b) H is abelian

(c) index of H in Sn is 2 (d) H is cyclic

13. Let G be a cyclic group of order 60. Then the number of subgroup of G is

(a) 1 (b) 2 (c) 4 (d) 12

14. Which of the following is correct?

(a) An infinite group can have unique element of finite order.

(b) ∃ an infinte order group in which order of every element is finite.

(c) For every infinite order a group G, ∃ a ∈ G such that O(a) is finite.

(d) None of these

15. Select the correct statement

(a) Every finite group has abelian subgroup

(b) Every finite group has cyclic subgroup

(c) Every finite group has a self inverse element other than identity

(d) none of the above

16. In the group (ℤ, +) the subgroup generated by 2 and 7 is

(a) ℤ (b) 5ℤ (c) 9ℤ (d) 14 ℤ

17. The number of elements of order 5 in the symmetric group S5 is

(a) 5 (b) 20 (c) 24 (d) 12

18. If H be a subgroup of ℤ30, then which of the following pair can be together in H

(a) {2, 28} (b) {8, 22} (c) {14, 16} (d) none of these

19. G = (P(ℕ), ∆)

H = {X ∈ P(ℕ) | |X| = finite} then

(a) H is not subgroup (b) H is non cyclic subgroup of G

(c) Every element of H is finite order (d) None of these

20. In a group, order of an element a is 12, then order of a5 is

(a) 12 (b) 10 (c) 14 (d) 8


(Group & Subgroup)

1. Let (ℂ∗,⋅) be the group of all non zero complex numbers under multiplication then

(a) (ℂ∗,⋅) has infinite elements of finite order.

(b) z = reiθ is of finite order if θ is rational multiple of 2π

(c) If O(H) = n, H < ℂ∗ then converse of Lagrangeʼs theorem hold in H

(d) If O(H) = 100, H < C∗ then ∃ a ∈ ℂ∗ between a ∈ H and a3 = e

2. Let H be a subgroup of a group G, consider the statement

(I) Every coset of H is a subgroup of G

(II) At least one coset of H is a subgroup of G then

(a) I true, II false (b) II true, I false

(c) I and II both are true (d) Neither I nor II are true

3. Choose the correct statements

(a) An abelian group with two elements of order 2 must have a subgroup of order 4.

(b) Every permutation of even order is odd permutation

(c) Let H ⊆ Sn such that H = {σ ∈ Sn ∶ Sign(σ) = 2} is a subgroup of Sn

(d) ∃ H < A4 such that O(H) = 6 because O(A4) = 12

4. Number of elements of order 2 in S6 is

(a) 70 (b) 75 (c) 80 (d) None of these

5. Let A = [1 10 1

] ∈ GL(2, ℤp) and H < GL(2, ℤp) such that H = ⟨A⟩ then order of H is

(a) 1 (b) 8 (c) ∞ (d) none of these

6. Consider S9, What is the maximum possible order of σ in S9

(a) 21 (b) 20 (c) 30 (d) 14

7. Which of the following is are⁄ true

(a) In a cyclic group, if ∃ an element of order k, then ∃ exactly ϕ(k) elements of order k.

(b) Let G = {eiθ: 0 ≤ θ ≤ 2π} be the group under multiplication then G has infinte elements of infinite

order

(c) G = {eiθ: 0 ≤ θ ≤ 2π} has infinite elements of finite order

(d) none of these

8. Let H = GL2(ℝ) be defined as H = {[1 a0 1

] ∶ a ∈ ℝ}

(a) H is non abelian group (b) H is not a subgroup of GL2(ℝ)

(c) ∃ an non identity element of H having finite order

(d) None of these

9. G = (ℂ∗,⋅), H = {z ∈ g ∶ O(z) = finite}

(a) H is not a subgroup of G (b) H is abelian subgroup of G

(c) H is cyclic subgroup of G (d) H is subgroup but non abelian

10. Let α, β ∈ Sn then f = β−1αβα−1 ∈ Sn is

(a) even only if α, β even (b) even only if α, β odd

(c) even if α odd and β even (d) always even

11. Let ℤ5 × S3 and S = {a ∈ G | O(a) = finite} then |S| is

(a) 9 (b) 8 (c) 10 (d) 7

12. Let G be the group of pnth roots of unity.

(a) Every proper subgroup of G is cyclic. (b) Every proper subgroup of G is of finite order.

(c) If p = 3, then there exists exactly 4 elements of order 8

(d) None of these

13. In Sn, let α be an r cycle, β be an s cycle and γ be an t cycle then

(a) αβ is even ⇔ r + s even (b) α β is even ⇔ r + s + 2 even

(c) αβγ is even ⇔ r + s + t even (d) αβγ is even ⇔ r + s + t odd

14. Let G = S3 × S3 then G has

(a) 12 elements of order 6 (b) 15 elements of order 2

(c) 10 elements of order 2 (d) 8 elements of order 3

15. Let ℂ∗ denote the multiplicative group of non zero complex numbers. Let G1 be the cyclic subgroup

generated by 1 + i and G2 be the cyclic subgroup generated by 1 + i

√2

(a) both G1 and G2 are infinite groups (b) G1 is finite but G2 is infinite group

(c) G2 is finite but G1 is infinite group (d) both G1 and G2 are finite groups

16. Let the group G = (ℝ, +) and the group (H = ℝ+,⋅)

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

(c) Neither G nor H cyclic (d) both G and H are cyclic

17. Which of the following numbers can be orders of permutations σ of 11 symbols such that σ does not fix

and symbol?

(a) 18 (b) 30 (c) 15 (d) 28

18. Let G = ℤ10 × ℤ15 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 10.

19. Which of the following group has a proper subgroup that is not cyclic?

(a) ℤ15 × ℤ17 (b) S3 (c) (ℤ, +) (d) (ℚ, +)

20. Let G1 be an abelian group of order 6 and G2 = S3. For J = 1,2; let PJ be the statement

GJ has a unique subgroup of order 2

(a) both P1 and P2 hold (b) neither P1 nor P2 hold

(c) P1 holds but not P2 (d) P2 holds but not P1


(Group and Subgroup)

1. Let p be a prime number let G be the group of all 2 × 2 matrices over ℤp with determinant 1 under

matrix multiplication then order of G is

(a) (p − 1)p(p + 1) (b) p2(p − 1)

(c) p3 (c) p2(p − 1) + p

2. Which of the following is are⁄ true

(a) ℤ2 ⊕ ℤ3 is isomorphic to ℤ6. (b) ℤ2 ⊕ ℤ3 is isomorphic to ℤ9

(c) ℤ4 ⊕ ℤ6 is isomorphic to ℤ24 (d) ℤ2 ⊕ ℤ3 ⊕ ℤ5 is isomorphic to ℤ30

3. Let S3 be the group of permutation of three distinct symbols. The direct sum S3 ⊕ S3 has an element of

order

(a) 2 (b) 6 (c) 9 (d) 18

4. Let H < Sn such that O(H) = 21 then

(a) H may contains 20 even and one odd permutation

(b) H may contain only even permutation

(c) H may contain only odd permutation

(d) H may contain 20 odd and one even permutation

5. The order of the element (2̅, 2̅) in ℤ4 × ℤ6 is

(a) 2 (b) 4 (c) 6 (d) 12

6. Let S3 be the group of all permutations on three symbols with identity element e, then the number of

elements in S3 that satisfy the equation x2 = e is

(a) 1 (b) 2 (c) 3 (d) 4

7. A group G is generated by the elements x, y with the relations x3 = y2 = (xy)2 = 1 then O(G) is

(a) 4 (b) 6 (c) 8 (d) 12

8. In the alternating group A6 number of elements of order 5 are

(a) 144 (b) 148 (c) 121 (d) 169

9. Let Dn = ⟨r, s: rn = s2 = e, rs = sr−1⟩ and Sn the group of all bijections on n symbols then

(a) Sn has a cyclic subgroup of order k ∀ k ≤ n (b) If x ∈ Dn such that x ∉ ⟨r⟩ then rx = xr−1

(c) ∃ x ∈ Dn such that x ∉ ⟨r⟩ then O(r) ≠ 2 (d) If n ≥ 3 is odd and x ∈ ℤ(Dn) then x = e

10. What is the cardinality of the set

{z ∈ ℂ ∶ z98 = 1, zn ≠ 1, for any 0 < n < 98}

(a) 0 (b) 12 (c) 42 (d) 49

11. Number of elements of order 4 in D10 is


12. Let Sn be the group of permutations on n symbols and f, g ∈ Sn

P ∶ If p and q have same syclic decomposition

Q ∶ If O(p) = O(Q) ⇒ p and q have same cyclic decomposition

(a) P true and Q false (b) P false and Q true

(c) P and Q both false (d) P and Q both true

13. Let G = ℤ × ℤm be a group then

(a) G is cyclic ∀ m ∈ ℕ (b) ∃m ∈ ℕ such that G is cyclic

(c) G is cyclic group of infinte order (d) none of these

14. Let H < Sn such that O(H) = 7 then

(a) H is also subgroup of An (b) n ≥ 7

(c) every subgroup K of Snsuch that K ≅ H ⇒ K is subgroup of An

(d) none of these

15. Number of subgroups of order 10 in ℤ100 × ℤ25 is

(a) 6 (b) 3 (c) 12 (d) 24

16. The minimal number of generators of D8 is

(a) 1 (b) 2 (c) 4 (d) 8

17. How many proper subgroups does the group ℤ ⊕ ℤ have?

(a) 1 (b) 2 (c) 3 (d) infinitely many

18. The order of the smallest possible non trivial group containing elements x and y such that

x7 = y2 = e and yx = x4y is

(a) 1 (b) 2 (c) 7 (d) 14

19. The number of elements of order 8 in the group ℤ24 is

(a) 1 (b) 2 (c) 3 (d) 4

20. Let G be the set of all 2 × 2 symmetric invertible matrices with real entries. Then with

multiplication G is

(a) An infinite group (b) a finite group (c) not a group (d) an abelian group


(Class Equation, Normal Subgroup, Factor Group, Homomorphism)

1. Let G be a group of order 7 and ϕ(x) = x4, x ∈ G then ϕ is

(a) not one one (b) not onto

(c) not a homomorphism (d) one one, onto and a homomorphism

2. The number of group of homomorphisms from the cyclic group ℤ4to the cyclic group ℤ7 is

(a) 7 (b) 3 (c) 2 (d) 1

3. Let G be a finite group then

(a) G is abelian if O(G) = pg where p and q distinct primes

(b) G is abelian if every non identiy element of G is of order 2

(c) G is abelian if the quotient group G

Z(G) is cyclic, Z(G) is centre of G

4. Let G be a cyclic group of order 12. Then the number of nonn isomorphic subgroups of G is

5. Suppose N is a normal subgroup of G which one of the following is true?

(a) If G is infinite group then G

N is an infinte group.

(b) If G is non abelian group then G

N is a non abelian group.

(c) If G is cyclic then G

N is an abelian group.

(d) If G is abelian group then G

N is a cyclic.

6. The number of distinct normal subgroups of S3 is ⋯

7. Let G be a group of order 17. The total number of non isomorphic subgroup of G is

(a) 1 (b) 2 (c) 3 (d) 17

8. Let G be a cyclic group of order 24. The total number of group isomorphisms of G onto itself is

(a) 7 (b) 8 (c) 17 (d) 24

9. Which of the following conditions on a group G implies that G is abelian?

(a) The order of G is p3 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 → G, defined by

f(x) = x−1 ∀ x ∈ G is homomorphism

10. Consider the quotiont group ℚ

ℤ of the additive group of rational numbers. The order of the element

2

3+ ℤ in

ℚ

ℤ is

(a) 2 (b) 3 (c) 5 (d) 6

11. Let G denote the group of all 2 × 2 matrix which are invertible with real entries

H1 = {A ∈ G ∶ det A = 1}

H2 = {A ∈ G ∶ A is upper triangular } then

(a) H1 is normal but H2 is not (b) H2 is normal but H1 is not

(c) H1 and H2 both normal (d) H1 and H2 both are not normal

12. Consider the following pairs

P ∶ℝ

ℤ and S1, (S1 = {z ∈ ℂ ∶ |z| = 1},⋅) are isomorphic to each other

Q ∶ (ℤ, +) and (ℚ, +) are isomorphic to each other

(a) both P and Q are true (b) P is true and Q is false

(c) P is false and Q is true (d) both P and Q are false

13. Let G be a finite group and H be the normal subgroup of G of order 2. Then order of the centre of G is

(a) 0 (b) 1

(c) an even integer ≥ 2 (d) an odd integer ≥ 3

14. Let G be the set of all 3 × 3 matrices (Real) such that MMt = MtM = I3 and

H = {M ∈ G ∶ |M| = 1}, then

(a) G is a group under matrix multiplication (b) H is a normal subgroup of G

(c) ϕ ∶ G → {1, −1} given by ϕ(M) = ⌊M⌋ is onto (d) G

H is abelian

15. Let G be a non abelian group of order 125. Then the total number of element in

Z(G) = {x ∈ G ∶ xg = gx for all g ∈ G} is ⋯

16. Let S9 be the permutation group on 9 symbols. The total number of elements of S9 commuting with

τ = (1 2 3)(4 5 6 7) in S9 is ⋯

17. Consider the following statements P and Q

P ∶ If H is a normal subgroup of order 4 of the symmetric group S4, thenS4

H is abelian.

Q ∶ If Q = {±1, ±i, ±j, ±k} is the quaterian group, then Q

{1, −1} is an abelian

(a) both P and Q are true (b) both P and Q are false

(c) only P is true (d) only Q is true

18. Let G = {x, y ∶ x5 = y2 = e | x2y = yx}. Then G is isomorphic to

(a) ℤ5 (b) ℤ10 (c) ℤ2 (d) ℤ3

19. G = {e, x, x2, x3, y, xy, x2y, x3y} with O(x) = 4, O(y) = 2, xy = yx3, then the number of elements in the

centre of the group G is

(a) 1 (b) 2 (c) 4 (d) 8

20. The number of non isomorphic groups of order 10 is ⋯


(Class equation, Normal subgroup, Factor group, Homomorphism)

1. Consider the group homomorphism

ϕ ∶ M2(ℝ) → ℝ given by ϕ(A) = trace (A)

The kernel of ϕ is isomorphic to which one of the following group?

(a)M2(ℝ)

H, {A ∈ M2(ℝ) ∶ ϕ(A) = 0} = H (b) ℝ2

(c) ℝ3 (d) GL(2, ℝ)

2. The number of group homomorphism from ℤ3 to ℤ9 is ⋯

3. Which of the following groups contains a unique normal subgroup of order 4?

(a) ℤ2 ⊕ ℤ4 (b) D4

(c) ℚ8 (d) ℤ2 ⊕ ℤ2 ⊕ ℤ2

4. Let ω = cos2π

3+ i sin

2π

3, M = (

0 ii 0

) , N = (ω 00 ω2) and G = ⟨M, N⟩ be the group generated by

matrices M and H under matrix multiplication. Then

(a) G

Z(G)≅ ℤ6 (b)

G

Z(G)≅ S3

(c) G

Z(G)≅ ℤ2 (d)

G

Z(G)≅ ℤ4

5. The number of elements in the conjugacy class of the 3 cyclic, σ = (2 3 4) in S6 is

(a) 20 (b) 40 (c) 120 (d) 216

6. Which one of the following group is simple

(a) S3 (b) GL(2, ℝ) (c) ℤ2 × ℤ2 (d) A5

7. Let G be the group of all symmetries of the square. Then the number of conjugate classes in G is

(a) 4 (b) 5 (c) 6 (d) 7

8. If Z(G) denote the centre of a group G, then the order of the quotient group G

Z(G) cannot be

(a) 4 (b) 6 (c) 15 (d) 25

9. Let Aut (G) denote the group of automorphism of a group G. Which one of the following is not a cyclic group?

(a) Aut (ℤ4) (b) Aut (ℤ6) (c) Aut (ℤ8) (d) Aut (ℤ10)

10. Let G = ℝ∗ = ℝ\{0} and H = {−1,1} be groups under multiplication. Then the map ϕ ∶ G → H

defined by ϕ(x) =x

|x| is

(a) not a homomorphism

(b) one one homomorphism which is not onto

(c) An onto homomorphism which is not one one

(d) An isomorphism

11. Consider ℤ5 and ℤ20 as group under modulo 5 and 20, then the number of homomorphism

ϕ ∶ ℤ5 → ℤ20 is

(a) 1 (b) 2 (c) 4 (d) 5

12. Let G be a cyclic group of order 8 then its group of automorphism has order

(a) 2 (b) 4 (c) 6 (d) 8

13. G = ⟨a, b ∶ a4 = b2 = 1, ba = a−1b⟩ if

Z(G) denote the centre of G, thenG

Z(G) is isomorphic to

(a) the trivial group (b) ℤ2 (c) ℤ2 × ℤ2 (d) ℤ4

14. Let S10 denote the group of permutation on ten symbols {1, 2, ⋯ , 10}. The number of elements of S10

commuting with σ = (1 3 5 7 9) is

(a) 5! (b) 5 ⋅ 5! (c) 5! 5! (d) 10!

5!

15. The cardinality of the centre of ℤ2 is

(a) 1 (b) 2 (c) 3 (d) 12

16. Let G and H be two groups. The groups G × H and H × G are isomorphic

(a) for any G and any H (b) only if one of them is cyclic

(c) only if one of them is abelian (d) only if G and H are isomorphic

17. H = ℤ2 × ℤ6, k = ℤ3 × ℤ4 then

(a) H is isomorphic to K since both are cyclic

(b) H is not isomorphic to k since 2 divides b and gcd(3,4) = 1

(c) H is not isomorphic to K since K is cyclic where as H is non cyclic

(d) H is not isomorphic to K since there is no homomorhism from H to K

18. The number of groups of order n upto isomorphism is

(a) finte for all values of n (b) finite only for finitely many values of n.

(c) finite for infinitely many values of n (d) infinite for some values of n

19. Let σ = (1 2 3 4 5 6 7 8 9

1 3 5 7 9 6 4 8 2)

τ = (1 2 3 4 5 6 7 8 9

7 8 3 4 9 6 5 2 1)

(a) σ and τ generate the group of permutation on {1, 2, ⋯ 9}

(b) σ is contained in the group generated by τ

(c) τ is contained in the geoup generated by σ

(d) σ and τ are in the same conjugacy class

20. The number of automorphisms of (ℤ, +) is

(a) 1 (b) 2 (c) 3 (d) 4


(Class equation, Normal subgroup, factor group, Homomorphism)

1. H = {e, (1 2)(3 4)}

K = {e, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} be subgroups of S4 then

(a) H and K are normal in S4. (b) H is normal in K and K is normal in A4

(c) H is normal in A4 but not normal in S4 (d) K is normal in S4 but H is not

2. Let G be a group of order 77. Then the centre of G is isomorphic to

(a) ℤ1 (b) ℤ7 (c) ℤ11 (d) ℤ77

3. The number of group homomorphism from the symmetric group S3 toℤ

6ℤ is

(a) 1 (b) 2 (c) 3 (d)

4. For any group of order 36 and any subgroup H of G of order 4

(a) H ⊂ Z(G) (b) H = Z(G)

(c) H is normal in G (d) H is an abelian group

5. Consider the group G =ℚ

ℤ. Let ne be a positive integer. Then these is a cyclic subgroup of order n

(a) not necessarily (b) yes, uniue one

(c) yes but need not be unique (d) never

6. The group S3 ⊕ℤ

2ℤ is isomorphic to

(a)ℤ

12ℤ (b)

ℤ

6ℤ⊕

ℤ

2ℤ

(c) A4, the alternation group of order 12 (d) D6, the dihedral group of order 12

7. Let σ = (1 2)(3 4 5), τ = (1 2 3 4 5 6) be permutations in S6, the group of permutations on six

symbols. Which of the following statements are true?

(a) The subgroup ⟨σ⟩ and ⟨τ⟩ are isomorphic to each other.

(b) σ and τ are conjugate in S6

(c) ⟨σ⟩ ∩ ⟨τ⟩ is the trivial group (d) σ and τ commute

8. Determine which of the following cannot be the class equation of a group.

(a) 10 = 1 + 1 + 1 + 2 + 5 (b) 4 = 1 + 1 + 2

(c) 8 = 1 + 1 + 3 + 3 (d) 6 = 1 + 2 + 3

9. The total number of non isomorphic groups of order 122 is

(a) 2 (b) 1 (c) 61 (d) 4

10. Let G be a non abelian group. Then its order can be

(a) 25 (b) 55 (c) 125 (d) 35

11. The number of conjugacy classes in the permutation group S6 is

(a) 12 (b) 1 (c) 10 (d) 6

12. Which of the following cannot be the class equation of a group of order 10?

(a) 10 = 1 + 1 + 1 + 2 + 5 (b) 10 = 1 + 2 + 3 + 4

(c) 10 = 1 + 2 + 2 + 5 (d) 10 = 1 + 1 + 2 + 2 + 2 + 2

13. Upto isomorphism, the number of abelian groups of order 108 is

(a) 12 (b) 9 (c) 6 (d) 5

14. Let G be a finite abelian group of order n. Pick the correct statement from the below.

(a) If d

n, there a subgroup of G of order d. (b) If

d

n, there exists an element of order d in G.

(c) If every proper subgroup of G is cyclic then G is cyclic.

(d) If H is a subgroup of G, there exists a subgroup N of G such that G

N≅ H.

15. Consider the following subset of the group of 2 × 2 non singular matrices over ℝ.

G = {(a b0 d

) : a, b, d ∈ ℝ, ad = 1 } and H = {(1 b0 1

) : b ∈ ℝ} which of the following statements

are correct?

(a) G forms a group under matrix multiplication. (b) H is normal subgroup of G.

(c) The quotient group G

H is well defined and is isomorphic to a group of 2 × 2 diagonal matrices

(over ℝ) with determinant 1.

(d) The quotient group G

H is well defined and is abelian.

16. What is the number of non singular 3 × 3 matrices over F2 the finite field with two elements?

(a) 168 (b) 384 (c) 23 (d) 32

17. Choose the correct statement

(a) There exists a finite group which is not a subgroup of Sn, for any n ≥ 1.

(b) Every finite group is a subgroup of An for some n ≥ 1.

(c) Every finite group is a quotient of An for some n ≥ 1.

(d) No finite abelian group is a quotient of Sn for n ≥ 3.

18. Let G be a group of order 125. Which of the following statements are neccessarily true?

(a) G has non trivial abelian subgroup (b) The centre of G is a proper subgroup.

(c) Centre of G has order 5. (d) There is a subgroup of order 25.

19. For an integer n ≥ 2. Let Sn be the group of permutation on n letters and An the alternating group. Let

ℂ∗ be the group of non zero complex numbers under multiplication. Which of the following are correct?

(a) For every integer n ≥ 2 there is a non trivial homomorphism χ: Sn → ℂ∗

(b) For every integer n ≥ 2, there is a unique non trivial homomorphism χ ∶ Sn → ℂ∗.

(c) For every integer n ≥ 3, there is a non trivial homomorphism χ ∶ An → ℂ∗.

(d) For every integer n ≥ 5, there is no non trivial homomorphism χ: An → ℂ∗.

20. The number of elements of finite order in G =GL(n, ℝ)

SL(n, ℝ)

(a) 1 (b) 2 (c) more than 2 but finite (d) infinite


(Sylow’s Theorem and Application)

1. Let p be a prime number. The order of a p sylow subgroup of the group GL50(𝔽p) of invertible 50 × 50

matrices with entries from the finite field Fp, equals

(a) p50 (b) p125 (c) p1250 (d) p1225

2. Let G be the S4 × S3. Then

(a) a 2 sylow subgroup of G is normal (b) a 3 sylow subgroup of G is normal

(c) G has a non trivial normal subgroup (d) G has a normal subgroup of order 72.

3. For a positive integer n ≥ 4 and a prime number p ≤ n. Let Up,n denote the union of all p SSG of the

alternating group An on n letters. Also, let Kp,n denote the subgroup of An generated by Up,n and let

|Kp,n| denote the order of Kp,n then

(a) |K2,4| = 12 (b) |K2,4| = 4

(c) |K2,5| = 60 (d) |K3,5| = 30

4. Let G be the simple group of order 168. What is the number of subgroup of G of order 7?

(a) 1 (b) 7 (c) 8 (d) 28

5. How many normal subgroups does a non abelian group G of order 21 have other than the identity

subgroup {e} and G?

(a) 0 (b) 1 (c) 3 (d) 7

6. Let G be the group of order 45,

(a) G has an element of order 9 (b) G has a subgroup of order 9

(c) G has a normal subgroup of order 9 (d) G has a normal subgroup of order 5.

7. In the group of all invertible 4 × 4 matrices with entries in the field of 3 elements, any 3 sylow subgroup

has cardinality

(a) 3 (b) 81 (c) 243 (d) 729

8. Let G be a simple group of order 60. Then

(a) G has six sylow 5 subgroups (b) G has four sylow 3 subgroups

(c) G has cyclic subgroup of order 6 (d) G has unieque element of order 2

9. Consider the symmetric group S20 and its subgroup A20 consisting of all even permutation. Let H be a 7

sylow subgroup of A20. Pick each correct statement

(a) |H| = 49 (b) |H| must be cyclic

(c) H is normal subgroup of A20 (d) Any 7 sylow subgroup of S20 is a subset of A20.

10. Let G be a group of order 15. Then the number of sylow subgroup of G of order 3 is

(a) 0 (b) 1 (c) 3 (d) 5

11. Let G be a finite group of order 200 then the number of subgroups of G of order 25 is

(a) 1 (b) 4 (c) 5 (d) 10

12. Upto isomorphism the number of abelian groups of order 105 is

(a) 2 (b) 5 (c) 7 (d) 49

13. The number of 5 sylow subgroup of ℤ20 is

(a) 1 (b) 4 (c) 5 (d) 6

14. The number of 5 sylow subgroup in the group of order 45 is

(a) 1 (b) 2 (c) 3 (d) 4

15. Let G be a group of order 231. The number of elements of order 11 in G is

16. The number of non isomorphic abelian groups of order 24 is ⋯

17. Let G be a group of order 45. Let H be a 3 sylow subgroup of G and K be a 5 sylow subgroup of G. Then

(a) both H and K are normal in G (b) H is normal in G but K is not normal

(c) H is not normal but K is normal (d) both K and K are not normal in G

18. Which of the following is false?

(a) Any abelian group of order 27 is cyclic. (b) Any abelian group of order 14 is cyclic.

(c) Any abelian group of order 21 is cyclic (d) Any abelian group of order 30 is cyclic.

19. Every subgroup of order 74 in a group of order 148 is normal (T/F).

20. Let G be a group. Suppose |G| = p2q where p, q are distinct primes satisfying q ≢ 1 mod p, which of

the following is always true?

(a) G has more than one p sylow subgroup (b) G has a normal p sylow subgroup.

(c) The number of q sylow subgroup of G is divisible by p.

(d) G has unique q sylow subgroup


(Sylow’s Theorem and Application)

1. Let p be a prime number. If P is a p sylow subgroup of same finite group G then for every subgroup of H

of G, H ∩ P is a p sylow subgroup of H. (T H⁄ )

2. Show that there are atleast two non isomorphic groups of order 198. Who that is all those groups the

number of elements of order 11 is the same.

3. Let G = GL(2, Fp). Prove that there is a sylow p subgroup is of G whose normalizer NG(H) is the group

of all upper triangular matrices in G. Hence, prove that the number of sylow subgroup of G is 1 + p

4. Pick the correct statement.

(a) ∃ a group of order 44 with a subgroup isomorphic to ℤ2 ⊕ ℤ2

(b) ∃ a group of order 44 with a subgroup isomorphic to ℤ4.

(c) ∃ a group of order 44 with a subgroup isomorphic to ℤ2 ⊕ ℤ2 and a subgroup isomorphic to ℤ4.

(d) ∃ a group of order 44 with a subgroup isomorphic to ℤ2 ⊕ ℤ2 or to ℤ4.

5. If G = GL(n, Fp) then the order of q SSG in G, n ≥ is

(a) qn(n−1) (b) qn(n+1)

2 (c) qn(n+1) (d) qn(n−1)

2

6. If G = A20 then the order of 7 SSG in G is

(a) 47 (b) 49 (c) 51 (d) 45

7. If G = ℤ3 × ℤ15 then how many subgroups of order 9 in G

(a) 1 (b) 2 (c) 3 (d) 4

8. If O(G) = 35, then how many subgroups in G

(a) 2 (b) 3 (c) 4 (d) 5

9. G = S3 × S3 then G has

(a) A subgroup of order 9 (b) ∄ a subgroup of order 9

(c) A subgroup of order 9 and it is normal (d) ∃ a subgroup of order 9 and it is not normal.


(a) If O(G) = 14, the subgroup of order 7 is normal

(b) If O(G) = 39, the subgroup of order 13 is normal

(c) IF O(G) = 39, the subgroup of order 3 is normal

(d) None of these


(a) Any finite p group has non trivial centre. (b) A p group may or may not be abelian

(c) Let G be an abelian group of order n then for every divisor m of n, G has a subgroup of order m

(d) None of these

12. Which of the following is not a p group?

(a) O(G) = 36 (b) O(G) = 128 (c) O(G) = 21 (d) O(G) = 98

13. Which of the following order of a group is not simple?

(a) O(G) = 28 (b) O(G) = 56 (c) O(G) = 12 (d) O(G) = 60

14. G = S3 × S4, then

(a) 2 SSG of S3 × S4 is normal (b) 3 SSG of S3 × S4 is normal

(c) G has normal subgroup of order 72 (d) None of these

15. If G is a finite non abelian simple group and H ≤ G then Index (H) in G is

(a) ≥ 5 (b) ≥ 3 (c) ≥ 4 (d) ≥ 2

16. Let a group of order 28, then

(a) If 2 SSG is normal then G is abelian. (b) If 7 SSG in normal then G is abelian

(c) 7 SSG in not normal in G (d) 2 SSG is always normal in G

17. Let G be a group of order 121

(a) G must be cyclic. (b) G must have an element of order 11

(c) G must have an element of order 121 (d) G cannot have an element of order 11

18. Which of the following is are⁄ correct?

(a) A5 cannot have subgroup of order 30 (b) H∆G, K < G ⇒ HK < G

(c) O(G) = 30 if 2 SG is unieque then G is abelia (d) S6 cannot have a subgroup of order 30

19. If H and K are subgrops of G with indexes 3 and 5 in G, then the index of H ∩ K in G is

(a) 3 (b) 5 (c) a multiple of 15 (d) not more than 8

20. The number of mutually non isomorphic group of order 45 is

(a) 0 (b) 1 (c) 2 (d) 4

Assignment 1 Basic Notions

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