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

$$f : G\rightarrow G$$

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 $\mathbb{C}$, given by $z\longmapsto \overline{z}$, the complex conjugate, is an automorphism of $\mathbb{C}$ (see exercise 1 for this section).

Let $G$ be a given group. We denote by $Aut(G)$ the set of all automorphisms of $G$. This set is a group with respect to the binary operation of composition of mappings: For clearly $1_G \in Aut(G)$ and is the identity element. The associative law is true for mappings with respect to composition and if $f \in Aut(G)$ then $f^{- 1}$ exists and $f^{-1}\in Aut(G)$ since

$$\begin{array}{c} f^{-1}(ab)=f^{-1}(f f^{-1}(a)f f^{-1}(b))... ...^{-1}(a))f( f^{-1}(b)))\\ =f^{-1}(a)f^{-1}(b), \end{array}$$

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 $G$. Let $a \in G$, and consider the mapping $\phi_a: G \rightarrow G$ defined by $\phi_a(x) = axa^{-1}$. We contend that $\phi_a \in Aut(G)$. We leave it as an exercise to show $\phi_a$ is 1-1 and onto. We note $\phi_a(xy) = axya^{-1} = (axa^{-1}) (aya^{-1}) = \phi_a(x) \phi_a(y)$, and so $\phi_a$ is an automorphism of $G$. It is called the inner automorphism determined by $a$.

For future reference, we note here the following result.

Proposition 7.2.1   Let $G$ be a group, $H \leq G$, and $a \in G$. Then $aHa^{-1} \leq G$.

Proof: Since the inner automorphism $\phi_a: G \rightarrow G$ is a homomorphism, we can apply Theorem 7.1.4 to imply that the image of H under ķa, i.e., $\phi_a[H] = aHa^{-1}$, is a subgroup of $G$. In words, Proposition 7.2.1 says that the conjugate of a subgroup is a subgroup. $\Box$

All elements of $Aut(G)$ (if there are any) which are not inner automorphisms are called outer automorphisms. Let us denote the set of all inner automorphisms of $G$ by $Inn(G)$. We claim that $Inn(G)\lhd Aut(G)$. To show this, we consider the mapping $\psi$ of $G$ into $Aut(G)$ given by $a\longmapsto \phi_a$, i.e.,

$$\psi(a) = \phi_a .$$ (7.1)

It is obvious that $\phi_a$ is onto $Inn(G)$. Also $\psi(ab)\phi_{ab}$, but

$$\begin{array}{c} \phi_a\phi_(x)=\phi_a(\phi_b(x))\\ =\ph... ...{-1}a^{-1}\\ =(ab)x(ab)^{-1}\\ \phi_{ab}(x). \end{array}$$

Thus $\phi_{ab} = \phi_{a}\phi_{b}$ so the mapping $\psi$ preserves the operation, i.e., $\psi$ is a homomorphism. Theorem 7.1.4 implies that the image at $Inn(G) = \psi [G]$ is a subgroup of $Aut(G)$. Now let $f \in Aut(G)$. Then

$$\begin{array}{c} f\phi_a f^{-1}(x)=f(af^{-1}(x)a^{-1})\\ ... ...)^{-1})\\ =f(a)xf(a)^{-1}\\ =\phi_{f(a)}(x), \end{array}$$

i.e., $f\phi_a f^{-1} = \phi_{f(a)}\in Inn(G)$. Thus $Inn(G)\lhd Aut(G)$.

Finally, let us consider the kernel, $Ker(\psi )$, of the homomorphism given in (7.1). Let $K = Ker(\psi )$. Now $K$ consists of those and only those elements $a \in G$ such that $\phi_a = 1_G$, i.e., $\phi_a(x) = 1_G(x) = x$, for all $x\in G$. In other words,

$$\phi_a(x) = axa^{-1} = x,$$

for all $x\in G$. Thus $K = Z(G)$, the center of $G$. Thus the FHT (Theorem 7.1.8) implies that

$$Inn(G) \cong G/Z(G).$$

We have therefore established the following result.

Theorem 7.2.2   The set $Inn(G)$ of all inner automorphisms of a group $G$ is a normal subgroup of the group $Aut(G)$ of all automorphisms of $G$. Moreover, $Inn(G) \cong G/Z(G)$.