Homomorphisms

A function is a homomorphism if for all , in G, .

In this definition, note that the product of with on the left side of the equation takes place in G whereas the product with takes place in H.

An onto homomorphism is called an epimorphism while an injective or one to one homomorphism is called a monomorphism. A bijective homomorphism is an isomorphism. There are some other named types of homomorphism which we summarize in the following table.

Here is a summary of some of the properties of a homomorphism . Elements such as will be representative of elements of G and elements of H.

• 
• 
• . In particular, the images of a cyclic subgroup of G are determined by the image of a generator.
• If g has finite order so does and the order of divides the order of g. For a monomorphism, the orders of g and are equal.

Note that if then the image of under is determined by the images of x under for each . For example, if and then .

As regard the behavior with subgroups, we first define two important sets associated with a homomorphism. First, the kernel of a homomorphism is and is denoted by . Second, the set (or sometimes is used) is the image of G in H. It is of course the range of .

• is a normal subgroup of G
• is a subgroup of H. They are equal if and only if is an epimorphism.
• If , is the coset .
• If K is a subgroup of G then is a subgroup of (and hence a subgroup of H). Moreover if K is normal in G then is a normal subgroup of (although not necessarily a normal subgroup of H).
• If L is a subgroup of H then is a subgroup of G.

• Pointers in creating a homomorphism
• Natural homomorphism
• Isomorphism
• Fundamental Theorem of Isomorphism