Group Theory Quiz

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Question 1

Consider the set S = {1, ω, ω2}, where ω and w2 are cube roots of unity. If * denotes the multiplication operation, the structure (S, *) forms

  • A group

  • A ring

  • An integral domain

  • A field

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Question 2

For the composition table of a cyclic group shown below

 

Which one of the following choices is correct?

  • a, b are generators

  • b, c are generators

  • c, d are generators

  • d, a are generators

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Question 3

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

  • Commutativity

  • Associativity

  • Existence of inverse for every element

  • Existence of identity

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Question 4

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

Note - This question was Numerical Type.

  • 1

  • 5

  • 7

  • 35

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Question 5

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


Note - This was Numerical Type question.

  • 42

  • 84

  • 1

  • 28

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Question 6

Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L != G and that the size of L is at least 4. The size of L is __________.

  • 3

  • 5

  • 7

  • 9

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Question 7

The set {1, 2, 3, 5, 7, 8, 9} under multiplication modulo 10 is not a group. Given below are four possible reasons. Which one of them is false?

  • It is not closed

  • 2 does not have an inverse

  • 3 does not have an inverse

  • 8 does not have an inverse

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Question 8

Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the concatenation operator for strings

  • does not form a group

  • forms a non-commutative group

  • does not have a right identity element

  • forms a group if the empty string is removed from ∑*

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Question 9

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?


  • Both G and H are always cyclic

  • G may not be cyclic, but H is always cyclic

  • G is always cyclic, but H may not be cyclic

  • Both G and H may not be cyclic

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Question 10

Let Zn be the group of integers { 0, 1, 2, ..., n-1} with addition modulo n as the group operation. The number of elements in the group Z2 * Z3 * Z4 that are their own inverses is 

  • 3

  • 5

  • 4

  • 2

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There are 11 questions to complete.

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