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).

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