D3 - automorphism

• eg: $\varphi :C\to C$ given by $\varphi (a+bi)=a-bi$ is an automorphism of the group of complex numbers under addition

• let $G=SL(2,R)$, the group of $2\times 2$ matrices with determinant $1$, and $M$ be any $2\times 2$ matrix with determinant $1$
• ${\varphi }_M:G\to \overline{G}$ defined as ${\varphi }_M(A)=MA{M}^{-1}\mathrm{\forall }A\in G$

• ${\varphi }_M$ is called conjugation by $M$

image: J Gallian, Contemporary Abstract Algebra

• $\text{Aut}(G)$ denotes the set of all automorphisms of $G$
• $\text{Inn}(G)$ denotes all its inner automorphisms

• to compute: $\text{Aut}(Z_{10})$
• any $k\in {\mathbb{Z}}_{10}$ can be written as $k=\underset{k\text{ times}}{\underbrace{1+1+\cdots +1}}$
• for $\alpha \in \text{Aut}(Z_{10})$, $\alpha (k)=k\alpha (1)$, so need to simply find $\alpha (1)$
• $|\alpha (1)|=10$, so the candidates are $1,3,7,9$ as the other elements do not generate $Z_{10}$
• denote the mappings as: ${\alpha }_a(1)=ak(\text{mod }10)$, ie. four candidates are: ${\alpha }_1,{\alpha }_3,{\alpha }_7,{\alpha }_9$
• ${\alpha }_1(k)=k$ is the identity
• the others are also automorphisms as $\text{mod }10$ preserves addition and is reversible
  • $x\mathrm{mod}10=y\mathrm{mod}10⟹3x\mathrm{mod}10=3y\mathrm{mod}10$
  • ${\alpha }_3(a+b)=3a+3b={\alpha }_3(a)+{\alpha }_3(b)$

• $({\alpha }_3{\alpha }_3)(1)={\alpha }_3(3)=9={\alpha }_9(1)⟹{\alpha }_3^2={\alpha }_9$
• similarly, ${\alpha }_3^3={\alpha }_7$, and ${\alpha }_3^4={\alpha }_1$
• thus, it is cyclic
• $\text{Aut}{Z}_{10}\cong U(10)$