# 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 1 Explanation:
 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 2 Explanation:

 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 3 Explanation:
 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 4 Explanation:
 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 5 Explanation:

There are 5 questions to complete.