For prime $p$, $\mathrm{\mid }\mathrm{Aut}((\mathbb{Z}\mathrm{/}p)\times (\mathbb{Z}\mathrm{/}p))\mathrm{\mid }=(p^2-1)(p^2-p)$

Given: The hypotheses and objects in the Statement.

An isomorphism is a bijective group homomorphism; an automorphism is an isomorphism from a group to itself, and $\mathrm{Aut}(G):=\{f:G\to G:f\text{ is an automorphism}\}\mathrm{.}$ (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$).

Let $G$ and $H$ be groups. Their external direct product has underlying set $G\times H:=\{(g,h):g\in G,h\in H\}$ and componentwise operation $(g,h)(g^{\mathrm{\prime }},h^{\mathrm{\prime }}):=(g{g}^{\mathrm{\prime }},h{h}^{\mathrm{\prime }})\mathrm{.}$ The fact that this operation makes $G\times H$ a group, with the indicated identity and inverses, is proved in thm-external-direct-product-is-a-group. Until that result is used, this definition introduces only the set and its componentwise binary operation. (The external direct product $G\times H$ with componentwise multiplication).

For groups $G$ and $H$, the componentwise operation of def-external-direct-product-of-groups makes $G\times H$ a group. Its identity is $(e_G,e_H)$, and $(g,h{)}^{-1}=(g^{-1},h^{-1})\mathrm{.}$ Moreover the coordinate maps ${\pi }_G(g,h)=g$ and ${\pi }_H(g,h)=h$ are group homomorphisms. ($G\times H$ is a group with identity $(e_G,e_H)$, coordinatewise inverses, and homomorphic coordinate projections).

For every prime $p$, the operations of addition and multiplication on $\mathbb{Z}\mathrm{/}p$ make it a field (def-field). (For every prime $p$, the two operations on $\mathbb{Z}\mathrm{/}p$ make it a field).

Let $n$ be a positive integer. Every class in $\mathbb{Z}\mathrm{/}n$ (def-integers-modulo-n) contains exactly one integer $r$ with $0\le r<n$. Consequently the map $r⟼[r{]}_n(0\le r<n)$ is a bijection from the von Neumann natural $n$ to $\mathbb{Z}\mathrm{/}n$, and $\mathrm{\mid }\mathbb{Z}\mathrm{/}n\mathrm{\mid }=n$. This includes $n=1$, where the only representative is $0$. For $n=0$, the map $a\mapsto [a{]}_0$ is a bijection $\mathbb{Z}\to \mathbb{Z}\mathrm{/}0$. (For $n\ge 1$, every class in $\mathbb{Z}\mathrm{/}n$ has one representative $r$ with $0\le r<n$, so $\mid \mathbb{Z}\mathrm{/}n\mid =n$; while $\mathbb{Z}\mathrm{/}0$ is in bijection with $\mathbb{Z}$).

1. If $A$ and $B$ are finite then $A\times B$ is finite and $\mid A\times B\mid =\mid A\mid \cdot \mid B\mid$ (def-finite-cardinality). 2. Let $m\in \mathbb{N}$ and let $A_0,\dots ,A_{m-1}$ be finite sets. Write ${\prod }_{i<m}{A}_i:=\{f:f\text{ is a function with domain }m\text{ and }f(i)\in A_i\text{ for every }i<m\}\mathrm{.}$ Then ${\prod }_{i<m}{A}_i$ is finite and $\mid {\prod }_{i<m}{A}_i\mid ={\prod }_{i<m}\mid A_i\mid$, the right-hand product being the $\mathbb{N}$-valued one of def-nat-finite-sum-and-product. (The product rule: $\mid A\times B\mid =\mid A\mid \mid B\mid$, and $\mid {\prod }_{i<m}{A}_i\mid ={\prod }_{i<m}\mid A_i\mid$).

Let $A$ be a finite set (def-countable) and let $B\subseteq A$. Then: 1. $B$ is finite; 2. $\mid B\mid \le \mid A\mid$ (def-finite-cardinality); 3. $\mid B\mid =\mid A\mid$ if and only if $B=A$; 4. every injection $f:A\to A$ is a bijection, and every surjection $f:A\to A$ is a bijection. (A subset of a finite set is finite, with $\mid B\mid \le \mid A\mid$, and equality holds if and only if $B=A$).

For a subset $S$ of a group $G$, the generated subgroup is $⟨S⟩:=\bigcap \{H:H\le G\text{ and }S\subseteq H\},$ the smallest subgroup of $G$ containing $S$. (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).