Group Theory

Question 1
Which of the following statements is/are TRUE for a group G?
MSQ
A
If for all x,y \in G, (xy)^2=x^2y^2, then G is commutative.
B
If for all x \in G, x^2=1, then G is commutative. Here, 1 is the identity element of G.
C
If the order of G is 2 , then G is commutative.
D
If G is commutative, then a subgroup of G need not be commutative.
GATE CSE 2022   Discrete Mathematics
Question 2
Let G be a group of order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?
A
Both G and H are always cyclic.
B
G may not be cyclic, but H is always cyclic.
C
G is always cyclic, but H may not be cyclic.
D
Both G and H may not be cyclic.
GATE CSE 2021 SET-1   Discrete Mathematics
Question 3
Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is _______.
A
34
B
35
C
5
D
7
GATE CSE 2020   Discrete Mathematics
Question 4
(G,*) is an abelian group. Then
A
x=x^{-1} for any x belonging to G
B
x=x^{2} for any x belonging to G
C
(x * y)^{2}=x^{2} * y^{2} , for any x,y belonging to G
D
G is of finite order
ISRO CSE 2018   Discrete Mathematics
Question 5
Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is ________.
A
42
B
21
C
24
D
84
GATE CSE 2018   Discrete Mathematics
Question 6
f (G,.) is a group such that (ab)^{-1}=a^{-1}b^{-1},\forall a,b \in G, then G is a/an
A
Commutative semi group
B
Abelian group
C
Non-abelian group
D
None of these
ISRO CSE 2016   Discrete Mathematics
Question 7
There are two elements x,y in a group (G,*) such that every element in the group can be written as a product of some number of x's and y's in some order. It is known that
x * x = y * y = x * y *x * y = y* x * y *x = e
where e is the identity element. The maximum number of elements in such a group is _______.
A
3
B
4
C
5
D
6
GATE CSE 2014 SET-3   Discrete Mathematics
Question 8
Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L\neqG and that the size of L is at least 4. The size of L is _______.
A
4
B
5
C
6
D
7
GATE CSE 2014 SET-3   Discrete Mathematics
Question 9
The arithmetic mean of attendance of 49 students of class A is 40% and that of 53 students of class B is 35%. Then the percentage of arithmetic mean of attendance of class A and B is
A
27.20%
B
50.25%
C
51.13%
D
37.40%
ISRO CSE 2011   Discrete Mathematics
Question 10
Consider the set S = {1, \omega ,\omega ^{2}}, where \omega and \omega ^{2} are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms
A
A group
B
A ring
C
An integral domain
D
A field
GATE CSE 2010   Discrete Mathematics
There are 10 questions to complete.

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