Some results from group theory

Definition 4   Let $(G , .)$ be a (multiplicative) group with neutral element $e$ . A nonempty subset $H \subseteq G$ is a subgroup of $G$ if the following three statements hold

• $e \in H$ ,
• for every $x, y \in H$ we have $x, y \in H$ ,
• for every $x \in H$ the inverse $x^{-1}$ of $x$ belongs to $H$ .

Theorem 3   For every subgroup $H$ of the additive abelian group $({\mbox{${\mathbb{Z}}$}}, +)$ there exists an element $a \in {\mbox{${\mathbb{Z}}$}}$ such that $H$ is the set of the multiples of $a$ , that is $H = a {\mbox{${\mathbb{Z}}$}}$ .

Theorem 4   Let $G$ be a multiplicative group with neutral element $e$ . Let $x \in G$ an element and gr($x$ ) the subgroup of $H$ consisting of all powers of $x$ (including $e = x^0$ and $x^{-1}$ the inverse of $x$ ). Let ${\Theta}(x)$ be the order of gr($x$ ), that is the cardinality of gr($x$ ). Then two cases arise

• either gr($x$ ) is infinite and then the powers of $x$ are pairwise different and thus $H$ is isomorphic to ${\mbox{${\mathbb{Z}}$}}$ .
• or gr($x$ ) is finite and we have the following properties
  1. ${\Theta}(x)$ is the smallest integer $n$ such that $x^n = e$ ,
  2. $x^m = x^{m'}$ iff $m \equiv m' \mod{ \, {\Theta}(x)}$ ,
  3. $H$ is isomorphic to $({\mbox{${\mathbb{Z}}$}}/n{\mbox{${\mathbb{Z}}$}}) \setminus \{0\}$ where $n = {\Theta}(x)$ ,
  4. $H = \{ e, x, x^2, \ldots, x^{n-1} \}$ where $n = {\Theta}(x)$ .

Theorem 5 (Lagrange)   For every subgroup $H$ of the finite group $G$ , the order (that is the cardinality) of $H$ divides that of $G$ .