Modern Algebra : Isomorphisms

Group Homomorphisms

Definitions, examples, and proofs talking about group homomorphisms are given.

Homomorphisms

A homomorphism is a function between two groups. Its a way to compare two groups for structural similarities. Homomorphisms are a powerful tool

for studying and cataloging groups.

Isomorphisms

An isomorphism is a homomorphism that is also a bijection. If there is an isomorphism between two groups G and H, then they are equivalent and we

say they are “isomorphic.” The groups may look different from each other, but their group properties will be the same.

The Kernel of a Group Homomorphism

The kernel of a group homomorphism measures how far off it is from being one-to-one (an injection). Suppose you have a group homomorphism f:G ?

H. The kernel is the set of all elements in G which map to the identity element in H. It is a subgroup in G and it depends on f. Different

homomorphisms between G and H can give different kernels.