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


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