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Free download in PDF Group Theory Multiple Choice 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.
(1)
A non empty set A is termed as an algebraic structure ________
[A]
with respect to binary operation *
[B]
with respect to ternary operation ?
[C]
with respect to binary operation +
[D]
with respect to unary operation –
Answer: with respect to binary operation *
(2)
Condition for monoid is __________
[A]
(a+e)=a
[B]
(a*e)=(a+e)
[C]
a=(a*(a+e)
[D]
(a*e)=(e*a)=a
(3)
A group (M,*) is said to be abelian if ___________
[A]
(x+y)=(y+x)
[B]
(x*y)=(y*x)
[C]
(x+y)=x
[D]
(y*x)=(x+y)
(4)
Matrix multiplication is a/an ____ property.
[A]
Commutative
[B]
Associative
[C]
Additive
[D]
Disjunctive
(5)
How many properties can be held by a group?
(6)
{1, i, -i, -1} is _____
[A]
semigroup
[B]
subgroup
[C]
cyclic group
[D]
abelian group
(7)
The set of all real numbers under the usual multiplication operation is not a group since
[A]
multiplication is not a binary operation
[B]
multiplication is not associative
[C]
identity element does not exist
[D]
zero has no inverse
Answer: zero has no inverse
(8)
The inverse of - i in the multiplicative group, {1, - 1, i , - i} is
[A]
1
[B]
-1
[C]
i
[D]
-i
(9)
If (G, .) is a group, such that (ab)2 = a2 b2 ∀ a, b ∈ G, then G is a/an
[A]
commutative semi group
[B]
abelian group
[C]
non-abelian group
[D]
none of these
(10)
If A = (1, 2, 3, 4). Let ~= {(1, 2), (1, 3), (4, 2)}. Then ~ is
[A]
not anti-symmetric
[B]
transitive
[C]
reflexive
[D]
symmetric
(11)
If the binary operation * is deined on a set of ordered pairs of real numbers as
(a, b) * (c, d) = (ad + bc, bd)
and is associative, then
(1, 2) * (3, 5) * (3, 4) equals
[A]
(74,40)
[B]
(32,40)
[C]
(23,11)
[D]
(7,11)
(12)
Match the following
A. Groups I. Associativity
B. Semi groups II. Identity
C. Monoids III. Commutative
D. Abelian Groups IV Left inverse
Codes. A B C D
[A]
IV I II III
[B]
III I IV II
[C]
II III I IV
[D]
I II III IV
(13)
An algebraic structure ____ is called a semigroup.
[A]
(P, *)
[B]
(Q, +, *)
[C]
(P, +)
[D]
(+, *)
(14)
A monoid is called a group if _______
[A]
(a*a)=a=(a+c)
[B]
(a*c)=(a+c)
[C]
(a+c)=a
[D]
(a*c)=(c*a)=e
(15)
A cyclic group can be generated by a/an _____ element.
[A]
singular
[B]
non-singular
[C]
inverse
[D]
multiplicative
(16)
A cyclic group is always _________
[A]
abelian group
[B]
monoid
[C]
semigroup
[D]
subgroup
(17)
If (G, .) is a group such that (ab)- 1 = a-1b-1, ∀ a, b ∈ G, then G is a/an
[A]
commutative semi group
[B]
abelian group
[C]
non-abelian group
[D]
None of these
(18)
If (G, .) is a group such that a2 = e, ∀a ∈ G, then G is
[A]
semi group
[B]
abelian group
[C]
non-abelian group
[D]
none of these
(19)
The set of integers Z with the binary operation "*" defined as a*b =a +b+ 1 for a, b ∈ Z, is a group. The identity element of this group is
[A]
0
[B]
1
[C]
-1
[D]
12
(20)
In the group (G, .), the value of (a- 1 b)- 1 is
[A]
ab-1
[B]
b- 1a
[C]
a-1b
[D]
ba-1
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