Conditional Probability and Bayes’ Theorem MCQs (Page 12)

111

What is the order of the stabilizer subgroup Gx for an element x in a set X under a group action of G?

Moderate · 64% Accuracy

Concept: #GroupActionsAndSylowTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A The size of G multiplied by the size of the orbit of x B The order of the orbit of x C The index of the orbit of x in G D The size of G divided by the size of the orbit of x

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112

If G is a group of order 15, what is the number of Sylow 5-subgroups of G?

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Concept: #GroupActionsAndSylowTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A 1 B 15 C 5 D 3

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113

What does the Correspondence Theorem (or Lattice Isomorphism Theorem) describe?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A The relationship between subgroups of G containing N and subgroups of G/N B The relationship between normal subgroups and abelian groups C The relationship between the kernel and the image of a map D The relationship between homomorphisms and isomorphisms

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114

If f: G -> H is a homomorphism and G is an abelian group, what must be true about the image of f?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A The image must be a cyclic group B The image must be an abelian group C The image must be a finite group D The image must be a simple group

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115

Which of the following is true regarding the kernel of a homomorphism f: G -> H?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A It is always a subgroup of G B It is always a simple group C It is always a subgroup of H D It is always a cyclic group

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116

If G is a cyclic group of order n, how many distinct homomorphisms exist from G to the group of integers under addition?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A Infinite B One C n D Zero

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117

What is the image of a group homomorphism f: G -> H?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A A subgroup of H B A normal subgroup of G C The entire group H D A quotient group of G

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118

If f: G -> H is an injective group homomorphism, what can be concluded about G?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A G must be a finite group B G is isomorphic to H C G must be an abelian group D G is isomorphic to a subgroup of H

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119

An automorphism of a group G is defined as which of the following?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A A homomorphism from G to the group of integers B A surjective homomorphism from G to a different group H C An isomorphism from G to a subgroup of G D An isomorphism from G to itself

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120

According to the Third Isomorphism Theorem, if K and N are normal subgroups of G such that N is a subset of K, then (G/N)/(K/N) is isomorphic to which of the following?

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Concept: #HomomorphismsAndIsomorphismTheorems #AbstractAlgebra(GroupTheory) #Mathematics

A K/N B G/N C G/K D K/G

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