Monoids and Groups for GATE

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Question 1

Which of the following option is True?

A
The union of any two subgroups of G is also a subgroup of G.
B
The intersection of any two subgroups of G is also a subgroup of G.
C
Both (A) and (B)
D
None

Question 2

If (G, *) is a cyclic group of order 480 then find the number of generators in G?

A
64
B
128
C
256
D
512

Question 3

The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively:

A
3 and 13
B
2 and 11
C
4 and 13
D
8 and 14

Question 4

For the composition table of a cyclic group shown below:

[Tex]\begin{array}{|c|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c} & \textbf{d}\\\hline \textbf{a} & \text{a}& \text{b} & \text{c} & \text{d} \\\hline \textbf{b} & \text{b}& \text{a} & \text{d} &\text{c}\\\hline \textbf{c} & \text{c}& \text{d} & \text{b} & \text{a}\\\hline \textbf{d} & \text{d}& \text{c} & \text{a} & \text{b} \\\hline \end{array}[/Tex]

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

Question 5

Which one of the following is NOT necessarily a property of a Group?

A

Commutativity
B

Associativity
C

Existence of inverse for every element
D

Existence of identity

Question 6

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is _______.

Question 7

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is ________.

Question 8

How many non-isomorphic abelian groups of order 64 are there?

A
10
B
9
C
12
D
11

Question 9

Let G be a cyclic group of order 60. How many subgroups does G have?

A
10
B
12
C
11
D
15

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GATE CS

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