# Group Theory

 Question 1
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 1 Explanation:
 Question 2
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 2 Explanation:
 Question 3
(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 3 Explanation:
 Question 4
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 4 Explanation:
 Question 5
$f (G,.)$ 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 5 Explanation:
 Question 6
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 6 Explanation:
 Question 7
Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L$\neq$G 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 7 Explanation:
 Question 8
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 8 Explanation:
 Question 9
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
Question 9 Explanation:
 Question 10
For the composition table of a cyclic group shown below

Which one of the following choices is correct?
 A a, b are generators B b, c are generators C c, d are generators D d, a are generators
GATE CSE 2009   Discrete Mathematics
Question 10 Explanation:
There are 10 questions to complete.