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Algebraic Structures and Homomorphisms

Description: Algebraic Structures and Homomorphisms Quiz
Number of Questions: 15
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Tags: algebraic structures homomorphisms groups rings fields
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Which of the following is an example of an algebraic structure?

  1. Group

  2. Ring

  3. Field

  4. Vector Space

  5. All of the above

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Correct Option: E
Explanation:

An algebraic structure is a non-empty set equipped with one or more binary operations that combine elements of the set to form other elements of the set.

What is a homomorphism?

  1. A function that preserves the algebraic structure of a set

  2. A function that preserves the order of a set

  3. A function that preserves the distance between elements of a set

  4. A function that preserves the cardinality of a set

  5. None of the above

Show answer
Correct Option: A
Explanation:

A homomorphism is a function between two algebraic structures that preserves the operations of the structures.

Which of the following is an example of a group?

  1. The set of integers under addition

  2. The set of real numbers under multiplication

  3. The set of complex numbers under addition

  4. The set of quaternions under multiplication

  5. None of the above

Show answer
Correct Option: A
Explanation:

A group is a non-empty set equipped with a binary operation that combines elements of the set to form other elements of the set, and that satisfies the following properties: associativity, identity element, and inverse element.

What is a ring?

  1. A non-empty set equipped with two binary operations, addition and multiplication

  2. A non-empty set equipped with one binary operation, addition

  3. A non-empty set equipped with one binary operation, multiplication

  4. A non-empty set equipped with three binary operations, addition, multiplication, and subtraction

  5. None of the above

Show answer
Correct Option: A
Explanation:

A ring is a non-empty set equipped with two binary operations, addition and multiplication, that satisfy the following properties: associativity, commutativity of addition, distributivity of multiplication over addition, identity element for addition, and inverse element for addition.

Which of the following is an example of a field?

  1. The set of rational numbers

  2. The set of real numbers

  3. The set of complex numbers

  4. The set of quaternions

  5. None of the above

Show answer
Correct Option: C
Explanation:

A field is a non-empty set equipped with two binary operations, addition and multiplication, that satisfy the following properties: associativity, commutativity of addition, distributivity of multiplication over addition, identity element for addition, inverse element for addition, identity element for multiplication, and inverse element for multiplication.

What is the kernel of a homomorphism?

  1. The set of elements that are mapped to the identity element

  2. The set of elements that are mapped to the zero element

  3. The set of elements that are mapped to themselves

  4. The set of elements that are mapped to the inverse of the identity element

  5. None of the above

Show answer
Correct Option: A
Explanation:

The kernel of a homomorphism is the set of elements in the domain that are mapped to the identity element in the codomain.

Which of the following is an example of an isomorphism?

  1. A homomorphism that is one-to-one and onto

  2. A homomorphism that is one-to-one but not onto

  3. A homomorphism that is onto but not one-to-one

  4. A homomorphism that is neither one-to-one nor onto

  5. None of the above

Show answer
Correct Option: A
Explanation:

An isomorphism is a homomorphism that is both one-to-one and onto.

What is the first isomorphism theorem?

  1. The kernel of a homomorphism is a normal subgroup of the domain

  2. The image of a homomorphism is a subgroup of the codomain

  3. The quotient group of the domain by the kernel of a homomorphism is isomorphic to the image of the homomorphism

  4. All of the above

  5. None of the above

Show answer
Correct Option: D
Explanation:

The first isomorphism theorem is a fundamental result in group theory that relates the kernel, image, and quotient group of a homomorphism.

Which of the following is an example of a cyclic group?

  1. The set of integers under addition

  2. The set of real numbers under multiplication

  3. The set of complex numbers under addition

  4. The set of quaternions under multiplication

  5. None of the above

Show answer
Correct Option: A
Explanation:

A cyclic group is a group that is generated by a single element.

What is the order of an element in a group?

  1. The number of elements in the group

  2. The number of times the element appears in the group

  3. The smallest positive integer $n$ such that $a^n = e$, where $e$ is the identity element of the group

  4. The largest positive integer $n$ such that $a^n = e$, where $e$ is the identity element of the group

  5. None of the above

Show answer
Correct Option: C
Explanation:

The order of an element in a group is the smallest positive integer $n$ such that $a^n = e$, where $e$ is the identity element of the group.

Which of the following is an example of a direct product of groups?

  1. The set of integers under addition

  2. The set of real numbers under multiplication

  3. The set of complex numbers under addition

  4. The set of quaternions under multiplication

  5. The Cartesian product of two groups

Show answer
Correct Option: E
Explanation:

The direct product of two groups is the Cartesian product of the two groups equipped with the operation of component-wise multiplication.

What is the fundamental theorem of finite abelian groups?

  1. Every finite abelian group is a direct product of cyclic groups

  2. Every finite abelian group is a direct product of prime-power groups

  3. Every finite abelian group is a direct product of simple groups

  4. Every finite abelian group is a direct product of perfect groups

  5. None of the above

Show answer
Correct Option: A
Explanation:

The fundamental theorem of finite abelian groups states that every finite abelian group is a direct product of cyclic groups.

Which of the following is an example of a simple group?

  1. The cyclic group of order 5

  2. The dihedral group of order 8

  3. The alternating group of degree 5

  4. The symmetric group of degree 3

  5. None of the above

Show answer
Correct Option: C
Explanation:

A simple group is a group that has no normal subgroups other than the trivial subgroup and the group itself.

What is the Sylow theorem?

  1. Every finite group has a Sylow subgroup for every prime divisor of its order

  2. Every finite group has a Sylow subgroup for every prime power divisor of its order

  3. Every finite group has a Sylow subgroup for every divisor of its order

  4. Every finite group has a Sylow subgroup for every perfect divisor of its order

  5. None of the above

Show answer
Correct Option: A
Explanation:

The Sylow theorem states that every finite group has a Sylow subgroup for every prime divisor of its order.

Which of the following is an example of a perfect group?

  1. The cyclic group of order 5

  2. The dihedral group of order 8

  3. The alternating group of degree 5

  4. The symmetric group of degree 3

  5. None of the above

Show answer
Correct Option: C
Explanation:

A perfect group is a group in which every proper normal subgroup is trivial.

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