Understanding Homomorphisms and Kernels

What is a homomorphism between two groups?

• A function that maps elements from one group to another while preserving group operations

• A function that is always one-to-one

• A function that maps elements from one group to another without any specific properties

• A function that is always onto

What does a homomorphism send identity elements to?

• Random elements

• Identity elements

• Kernel elements

• Inverse elements

Why do homomorphisms send inverses to inverses?

• Because inverses are not important in group theory

• Because it is a property of homomorphisms to preserve group operations

• Because inverses are always mapped to identity elements

• Because inverses are mapped to random elements

What is the kernel of a homomorphism?

• A subset of the codomain group

• A measure of how a homomorphism fails to be one-to-one

• A random collection of elements

• A set of elements that map to random elements in the codomain

What is true about the kernel of a homomorphism?

• It is always empty

• It is a subgroup of the codomain group

• It is a subgroup of the domain group

• It contains only the identity element of the codomain