Cauchy's theorem for finite abelian groups

Given: The objects and hypotheses in the statement.

Let $P$ be a property of naturals such that for every $n \in \mathbb{N}$, if $P(m)$ holds for all $m < n$ then $P(n)$. Then $P(n)$ holds for all $n \in \mathbb{N}$. (At $n = 0$ the hypothesis is vacuous, so $P(0)$ is forced.) (Strong (complete) induction).

Let $G$ be a finite group and $H\le G$. Then $|G|=[G:H]\,|H|.$ Consequently, under the canonical embedding $\iota:\mathbb N\to\mathbb Z$, $|H|$ divides $|G|$. (Lagrange's theorem: $|G|=[G:H]|H|$ for every subgroup $H$ of a finite group $G$).

Let $N\mathrel{\trianglelefteq}G$. If $[G:N]$ is finite, then the quotient group $G/N$ is finite and $|G/N|=[G:N].$ In particular, if $G$ is finite, then $|G/N|=\frac{|G|}{|N|}.$ (If $[G:N]$ is finite then $|G/N|=[G:N]$; for finite $G$ this equals $|G|/|N|$).

If $G$ is abelian and $N\mathrel{\trianglelefteq}G$, then $G/N$ is abelian. (Every quotient group of an abelian group is abelian).

Let $G$ be a group and let $N\mathrel{\trianglelefteq}G$ be a normal subgroup (def-normal-subgroup). The quotient group, or factor group, $G/N$ has the left cosets $G/N:=\{gN:g\in G\}$ as its elements (def-coset, def-index), with product $(gN)(hN):=ghN.$ Independence of the chosen representatives is proved in thm-coset-multiplication-well-defined-iff-normal, and the group axioms are proved in thm-quotient-group-laws. (The quotient group $G/N$ and coset product $(gN)(hN)=ghN$).

Let $G$ be a finite group such that the positive integer $|G|$ is prime. Then every $g\ne e$ has order $|G|$, satisfies $\langle g\rangle=G$, and hence generates $G$. In particular, $G$ is cyclic. (A finite group of prime order is cyclic and every nonidentity element generates it).

Let $G$ be a group, $g \in G$, and let orders be as in def-order-in-a-group. Throughout, a natural number written where an integer is expected means its image under the embedding $\iota : \mathbb{N} \to \mathbb{Z}$ of lem-nat-embeds-int. Finite order. Suppose $\operatorname{ord}(g) = n$ with $n \in \mathbb{N}$, $n \ge 1$. Then: 1. for every $k \in \mathbb{Z}$, $g^{k} = e$ if and only if $k = qn$ for some $q \in \mathbb{Z}$, that is, if and only if $n \mid k$ (thm-division-algorithm-in-z); 2. the powers $g^{0}, g^{1}, \dots, g^{n-1}$ are pairwise distinct: if $i, j \in \mathbb{N}$ with $i < n$, $j < n$ and $g^{i} = g^{j}$, then $i = j$; 3. $\langle g \rangle = \{\, g^{s} : s \in \mathbb{N},\ s < n \,\}$ and $\langle g \rangle \approx n$; so $\langle g \rangle$ is finite with $|\langle g \rangle| = n = \operatorname{ord}(g)$. Infinite order. If $\operatorname{ord}(g) = \infty$ then for $j, k \in \mathbb{Z}$, $g^{j} = g^{k}$ implies $j = k$; so the integer powers of $g$ are pairwise distinct and $\langle g \rangle$ is not finite. (If $\operatorname{ord}(g) = n$ then $g^{k} = e$ iff $k$ is an integer multiple of $n$, the powers $g^{0}, \dots, g^{n-1}$ are distinct, and $\langle g \rangle$ has exactly $n$ elements; if $g$ has infinite order then $g^{j} = g^{k}$ only for $j = k$).

If $G=\langle g\rangle$ is cyclic, then exactly one of the following applies: - if $g$ has infinite order, $G\cong(\mathbb Z,+)$; - if $g$ has finite order $n$, necessarily $n\ge1$, then $G\cong(\mathbb Z/n,+)$. (Every cyclic group is isomorphic to $(\mathbb Z,+)$ or to $(\mathbb Z/n,+)$ for its finite order $n\ge1$).