Exercises

1. Prove that the map $z\longmapsto \overline{z}$, where $z = a +bi$, $\overline{z}=a-ib$ is an automorphism of $\mathbb{C}$ under $+$.

2. In the text, we did not show that $Aut(G)$ was closed with respect to composition, i.e., that composition is a binary operation on $Aut(G)$. Show it. (You may use the result of exercise 2 for Section 7.1.)

3. For an arbitrary group $G$, let $a \in G$ and define $\phi_a(x) = axa^{-1}$ for all $x\in G$. Show $\phi_a$ is a 1-1 and onto map of $G$ onto $G$. Finally show that $\phi_a$ preserves the group operation. This exercise shows that $\phi_a \in Aut(G)$ ($\phi_a$ is the inner automorphism determined by $a$.)

4. Show that if $G$ is a group with trivial center ($Z(G) = \{e\}$), then its group of automorphisms, $Aut(G)$, is also a group with trivial center.

   (HINT: Let $f \in Z (Aut(G))$. For any $x\in G$, let $\phi_x \in Inn(G)$. Then $f\phi_x = \phi_xf$ (Why?). Use this to show that for any $y\in G$, $x^{-1}f(x) \in C_G(f(y))$. Infer the result from this.)