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Free download in PDF Group Theory Objective Type Questions and Answers for competitive exams. These short objective type questions with answers are very important for Board exams as well as competitive exams. These short solved questions or quizzes are provided by Gkseries.
(21)
(Z,*) is a group with a*b = a+b+1 ∀ a, b ∈Z. The inverse of a is
[A]
0
[B]
-2
[C]
a-2
[D]
-a-2
(22)
Let G denoted the set of all n x n non-singular matrices with rational numbers as entries. Then under multiplication G is a/an
[A]
subgroup
[B]
finite abelian group
[C]
infinite, non abelian group
[D]
ininite, abelian
Answer: infinite, non abelian group
(23)
Let A be the set of all non-singular matrices over real numbers and let * be the matrix multiplication operator. Then
[A]
A is closed under * but < A, * > is not a semi group
[B]
< A, * > is a semi group but not a monoid
[C]
< A, * > is a monoid but not a group
[D]
< A, * > is a group but not an abelian group
Answer: < A, * > is a group but not an abelian group
(24)
{ - 3 n : n ε Z } IS an abelian group under
[A]
subtraction
[B]
division
[C]
multiplication
[D]
addition
(25)
If G = { 1, -1, ι, - ι } is group under multiplication, then inverse of - ι is
[A]
1
[B]
−1
[C]
ι
[D]
None of Above
(26)
A monoid is always a
[A]
A group
[B]
a commutative group
[C]
a non abelian group
[D]
groupoid
(27)
If a, b are elements of a group G, then (ba)-1 =
[A]
a-1 b-1
[B]
b-1 a-1
[C]
a-1 b
[D]
b-1 a
(28)
A monoid is always a
[A]
A group
[B]
a commutative group
[C]
a non abelian group
[D]
semi – group
(29)
This is an abelian group { – 3 n : n ε Z } under?
[A]
division
[B]
subtraction
[C]
addition
[D]
multiplication
(30)
What is the inverse of – ι If G = { 1, -1, ι, – ι } is group under multiplication?
[A]
−1
[B]
ι
[C]
1
[D]
None of Above
31
The monoid is a?
[A]
a non-abelian group
[B]
groupoid
[C]
A group
[D]
a commutative group
Answer: a non-abelian group
32
What is an inverse of – i in the multiplicative group if {1, – 1, i , – i} is?
[A]
-1
[B]
1
[C]
i
[D]
None of these
33
What is the value of (a- 1 b)- 1 is in the group (G, .)?
[A]
b- 1a
[B]
ab-1
[C]
ba-1
[D]
a-1b
34
What is the inverse of an if (Z,*) is a group with a*b = a+b+1 ∀ a, b ∈Z?
[A]
-2
[B]
0
[C]
-a-2
[D]
a-2
35
Which sentence is true?
[A]
Set of all matrices forms a group under multiplication
[B]
Set of all rational negative numbers forms a group under multiplication
[C]
Set of all non-singular matrices forms a group under multiplication
[D]
Both (b) and (c)
Answer: Set of all non-singular matrices forms a group under multiplication
36
Which statement is false?
[A]
The set of rational integers is an abelian group under addition
[B]
The set of rational numbers form an abelian group under multiplication
[C]
The set of rational numbers is an abelian group under addition
[D]
None of these
Answer: The set of rational numbers form an abelian group under multiplication
37
What is the identity element In the group G = {2, 4, 6, 8) under multiplication modulo 10?
38
Let (Z, *) be an algebraic structure, where Z is the set of integers and the operation * is defined by n * m = maximum (n, m). Which of the following statements is TRUE for (Z, *) ?
[A]
(Z, *) is a monoid
[B]
(Z, *) is an abelian group
[C]
(Z, *) is a group
[D]
None of these
39
Some group (G, 0) is known to be abelian. Then which one of the following is TRUE for G ?
[A]
g = g-1 for every g ∈ G
[B]
g = g2 for every g ∈ G
[C]
(g o h) 2 = g2o h2 for every g,h ∈ G
[D]
G is of finite order
Answer: (g o h) 2 = g2o h2 for every g,h ∈ G
40
(ba)-1 =_____ If a, b are elements of a group G?
[A]
b-1 a
[B]
a-1 b
[C]
b-1 a-1
[D]
a-1 b-1
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