Understanding Homomorphisms and Isomorphisms

What is a homomorphism?

• A function that is always bijective.

• A function that only maps elements from a group to itself.

• A function between two groups that preserves the group structure.

• A function that maps elements from one group to another without preserving structure.

What makes a homomorphism an isomorphism?

• It must map elements to themselves.

• It must be injective only.

• It must be both injective and surjective.

• It must be surjective only.

In the context of group theory, what does 'bijective' mean?

• The function is only injective.

• The function is both injective and surjective.

• The function is only surjective.

• The function is neither injective nor surjective.

Which function is used as a homomorphism between positive real numbers under multiplication and all real numbers under addition?

• Cosine function

• Sine function

• Logarithm function

• Exponential function

Why is the logarithm function considered an isomorphism in the given example?

• Because it is neither injective nor surjective.

• Because it is injective but not surjective.

• Because it is surjective but not injective.

• Because it is both injective and surjective.

What is the group of non-zero complex numbers under multiplication denoted by?

• R with a multiplication sign

• S1

• C with a multiplication sign

• C with a plus sign

Why is the homomorphism between non-zero complex numbers and complex numbers with absolute value of one not an isomorphism?

• Because it is not injective.

• Because it is not surjective.

• Because it is neither injective nor surjective.

• Because it maps elements to themselves.

What does the term 'isomorphic' imply about two groups?

• They are identical in every aspect.

• They cannot be compared.

• They have the same group structure.

• They have different group structures.

What does the prefix 'iso' in 'isomorphism' mean?

• Equal

• Opposite

• Similar

• Different

What is the significance of isomorphic groups in mathematics?

• They have different structures but look the same.

• They have the same structure but look different.

• They are always identical in every way.

• They cannot be compared.