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 = x⁰ 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 xⁿ = 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}$}$ where n = $\Theta$ (x),
4. H = {e, x, x²,..., x^n-1} where n = $\Theta$ (x).