MATHOPHILIC EDUCATOIN
Contact no: 7239082744, 7065194893
(GROUP THEORY)
ASSIGNMENT 1
(1) Let𝜎 be an element of permutation group s5 .then maximum possible order
of 𝜎 is ?
(a) 6
(b) 5
(c) 10
(d) 15
(2) In the permutation group sn(n≥5). If H is the smallest subgroup containing
all the 3 cycle , then which one of the following is true ?
(a) Order of H is 2
(b) H is abelian
(c) Index of H in sn is 2
(d) H=sn
(3) Let H denotes the group of all 2x2 invertibal matrices over z 5 under the
2 3
usual matrix multiplication . Then order of matrix [
] in H is ?
1 2
(4) Let s3 be group of permutations of three distinct symbols . The direct sum
s3⊕s3 has element of order ?
(a) 18
(b) 9
(c) 6
(d) 4
(5) Let H be a non abeliangroup . let 𝛼 ∈ G have order 4 and let 𝛽 ∈ G have
order 3 .Then order of element 𝛼𝛽 in G is
(a) 5
(b) 12
(c) Is of the form for k≥2
(d) Need not be finite
(6) let P be a prime number . let G be the group of all 2x2 matrices over Z p with
determinant 1 under matrix multiplication . then order of G is
(a) (p-1) (p) (p+1)
(b) P3
(c) P2(P-1)
(d) P2(P-1)+P
(7) Let G be finite group whose oder is not divisible by 3 , then
P: for every g ∈ G ,∋ an h ∈ G such that g=h3
Q:for every g ∈G ,∋an h ∈ G such that g=h2
(a) P and Q both are correct
(b) P is correct and Qis incorrect
(c) P is incooectand Q is correct
(d) P and Q are ibcorrect
(8) The number of elements of s5 which are their own inverse equals
(a) 10
(b) 11
(c) 25
(d) 26
(9) The number of nbelian subgroup of order 6 in K4xD8
(a) 0
(b) 1
(c) 10
(d) 2
(10)
Which is / are correct
(a) In a cyclic group , if ∃ an element of order K , then ∃ exactly ∅ (K)
elements of order K.
(b) Let G= { 𝑒 𝑖𝜃 | 0≤ 𝜃 ≤2𝜋 be group over mltipication than nG has
infinite elements of infinite order
(c) G {𝑒 𝑖𝜃 | 𝑜 ≤ 𝜃 ≤ 2𝜋} has infinite elements of finite order
(d)
None of these
11. Let H ⊆ GL(2,ℝ) be defined as H =
{ [10 𝑎1 ] | 𝑎 Є ℝ }
A). H is non abelian group
B). H is not a subgroup of GL(2, ℝ)
C). ∃ an identity element of H having finite order
D). none of these
12. G = ( ℂ* , . ) , H = { z Є G | o(z) = finite } then
A) H is not subgroup of G
B) H is abelian subgroup of G
C) H is cyclic subgroup of G
D) H is subgroup but not abelian
13. Which of the following groups has a proper subgroup that is not cyclic
A) Z15 X Z7
B) S3
C) (𝕫,+)
D) (ℚ, + )
14. Order of smallest possible non-trivial group containing elements x and y such
that xT = y2 = e and yx=x4y is
A) 1
B) 2
C) 7
D) 14
15. Let G1 be abelian group of order 6 and G2 = S3 , for j=1,2 Let Pj be statements
“ Gj has unique subgroup of order 2 “ then
A)
B)
C)
D)
P1 & P2 both holds
Neither P1 nor P2 holds
P1 holds not P2
P2 holds not P1
16. Then value of α for Which G = { α,1,2,3,9,19,20 } is cyclic group under
multiplication modulo 56 is
A) 5
B) 15
C) 25
D) 35
17. Let G be group of order 49 , then
A) G is abelian
B) G is cyclic
C) G is non abelian
D) Centre of G has order 7
18. There are no infinite group with Subgroup of index 5 [T/F]
19. A cyclic group of order 60 has
A) 15 generator
B) 12 generator
C) 16 generator
D) 20 generator
20. Every infinite abelian group has atleast one element of infinite order [T/F]
21. A group G is generated by the elements x,y with relations x 3 = y2 = (xy)2 = 1
order of G is
A) 4
B) 6
C) 8
D) 12
22. Let an denotes the number of those permutations 𝜎 on { 1,2,………,n} such that
𝜎 is product of exactly 2 disjoint cycles , then
A) a5 = 50
B) a4 = 14
C) a4 = 11
D) a5 = 40
23. Which of following can be orders of permutations 𝜎 of 11 symbols such that 𝜎
does not fix any symbols ?
A) 18
B) 30
C) 15
D) 28
24. Consider H = { f Є Sn | f(1) =1 , f(3) =3 } then
A) H is not subgroup of S5
B) H is non abelian subgroup of S5
C) H is abelian subgroup of S5
D) H is cyclic Subgroup of S5
25. Let G = Z X Zm be a group then
A) G is cyclic ∀ m Є ℕ
B) ∃ m Є ℕ such that G is cyclic
C) G is cyclic group of infinite order
D) none of these
26. Let H = { Z Є ℂ* : |z| = 1 } ⊆ ℂ* then
A) H is Subgroup of C*
B) H is Cyclic Subgroup of C*
C) H is abelian non cyclic Subgroup of C *
D) ∃ ∝ Є 𝐻 Such that o(∝) = infinite
27. Let G = S3 X 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
28. A =
[ 10 11 ] Є GL (2,Z ) and H < GL (2, Z ) Such that H = <A> then order of H
is
A) P
B) 8
C) ∞
D) none of these
p
p
29. Number of elements of order 4 in D10 is
A) 2
B) 4
C) 6
D) none of these
30. Number of non commutative binary operation of set with cardinality 2
A) 7
B) 8
C) 9
D) 10
31. Which of following is correct
A) (Z6 , +6 ) < (Z12 , +12 )
B) (Z6 , +6 ) ≮ (Z12 , +12 )
C) mZ is subgroup of Z , ∀ m Є ℕ
D) mZ is not subgroup of Z , ∀ m Є ℕ
32. In a group , order of element a is 12 , then order of a5 is
A) 12
B) 10
C) 14
D) 8
33. Order of a and x in a group are 3 and 4 , order of x -1ax is
A) 3
B) 4
C) 6
D) 12
34. Let ( ℂ* , . ) be group of all non-zero Complex number under multiplication
then
A) ( ℂ* , . ) has infinite elements of finite order
B) Z = r𝑒 𝑖𝜃 is of finite order if 𝜃 is rational multiple of 2𝜋
C) if o(H) = n , H < ℂ* then converse of lagrange theorem holds in H
D) if o(H) = 100 ; H < ℂ* then ∃ a Є ℂ* between a Є H & a3 = e
35. Let G be abelian group of order 10. Let S = { g Є G : g -1 = g } , then number of
non identity element in S is
A) 5
B) 2
C) 1
D) 0
36. The number of all Subgroups of group ( Z60 , +) of integer modulo 60 is
A) 2
B) 10
C) 12
D) 60
37. In the group ( Z ,+ ) , Subgroup generated by 2 and 7 is
A) Z
B) 5Z
C) 9Z
D) 14Z
38. Order of elements (2,2) in Z4 X Z6 is
A) 2
B) 6
C) 4
D) 12
39. Consider following statements
S : Every non-abelian group has a non-trivial abelian Subgroup
T : Every non-trivial abelian group has cyclic Subgroup then
A)
B)
C)
D)
Both S and T are False
S is True and T is False
T is True and S is False
Both S and T are True
40. Number of 2X2 matrices over Z3 with determinant 1
A) 24
B) 60
C) 20
D) 30
41. Suppose G is Cyclic group and , 𝜏 Є G are such that o(𝜎) = 12 and o(𝜏) = 21 ,
then order of smallest group containing 𝜎 and 𝜏 is _________________
42. Every finitr group of order 17 is abelian [T/F]
43. Every group of order 6 is abelian [T/F]
44. G = { Z Є ℂ | Zn = 1 for some positive integer } under multiplication of complex
numbers
A) G is group of finite order
B) G is group of infinite order , but every element of G has finite order
C) G is cyclic group
D) none of these
45. All Non trivial proper Subgroup of ( ℝ , + ) are cyclic [T/F]
46. How many proper Subgroups does the groups Z⨁ Z have ?
A) 1
B) 2
C) 3
D) infinitely many
47. For a group G ,Let F(G) denote the Collection of all Subgroup Of G , Which
Situation Can Occur ?
A) G is finite but f(G) is infinite
B) G is infinite but f(G) is finite
C) G is countable but f(G) is uncountable
D) G is uncountable but f(G) is countable
48. Let G be group , H and K be Subgroups of G . if both H and K have 12 elements
Which of following numbers cannot be cardinality of set HK = { hk : h Є H , k Є K } ?
A) 72
B) 60
C) 48
D) 36
49. Let G be cyclic group of order 6 . then numbers of elements g Є G such that G
= <g> is
A) 5
B) 3
C) 4
D) 2
50. if x,y and z are elements of a group such that xyz = 1 then
A) yzx = 1
B) yxz = 1
C) zxy = 1
D) zyx = 1