# Special Homomorphisms and Isomorphisms

We conclude this chapter with some considerations of special homomorphisms and isomorphisms. First we define the notion of an endomorphism. A homomorphism

of a group into itself is called an endomorphism. An isomorphism of a group G onto itself is called an automorphism. For example, the mapping of the additive group of complex numbers , given by , the complex conjugate, is an automorphism of (see exercise 1 for this section).

Let be a given group. We denote by the set of all automorphisms of . This set is a group with respect to the binary operation of composition of mappings: For clearly and is the identity element. The associative law is true for mappings with respect to composition and if then exists and since

since composition is represented by juxtaposition. Is this sufficient to show Aut(G) is a group with respect to composition? WHY or WHY NOT? (See exercise 3 for this section.)

Now we consider special kinds of automorphisms of a group . Let , and consider the mapping defined by . We contend that . We leave it as an exercise to show is 1-1 and onto. We note , and so is an automorphism of . It is called the inner automorphism determined by .

For future reference, we note here the following result.

Proposition 7.2.1   Let be a group, , and . Then .

Proof: Since the inner automorphism is a homomorphism, we can apply Theorem 7.1.4 to imply that the image of H under ía, i.e., , is a subgroup of . In words, Proposition 7.2.1 says that the conjugate of a subgroup is a subgroup.

All elements of (if there are any) which are not inner automorphisms are called outer automorphisms. Let us denote the set of all inner automorphisms of by . We claim that . To show this, we consider the mapping of into given by , i.e.,

 (7.1)

It is obvious that is onto . Also , but

Thus so the mapping preserves the operation, i.e., is a homomorphism. Theorem 7.1.4 implies that the image at is a subgroup of . Now let . Then

i.e., . Thus .

Finally, let us consider the kernel, , of the homomorphism given in (7.1). Let . Now consists of those and only those elements such that , i.e., , for all . In other words,

for all . Thus , the center of . Thus the FHT (Theorem 7.1.8) implies that

We have therefore established the following result.

Theorem 7.2.2   The set of all inner automorphisms of a group is a normal subgroup of the group of all automorphisms of . Moreover, .

Subsections

David Joyner 2007-08-06