Some results from group theory

Definition 4   Let (G,.) mathend000# be a (multiplicative) group with neutral element e mathend000#. A nonempty subset H $\subseteq$ G mathend000# is a subgroup of G mathend000# if the following three statements hold

• e $\in$ H mathend000#,
• for every x, y $\in$ H mathend000# we have x y $\in$ H mathend000#,
• for every x $\in$ H mathend000# the inverse x^-1 mathend000# of x mathend000# belongs to H mathend000#.

Theorem 3   For every subgroup H mathend000# of the additive abelian group ($\mbox{${\mathbb Z}$}$, +) mathend000# there exists an element a $\in$ $\mbox{${\mathbb Z}$}$ mathend000# such that H mathend000# is the set of the multiples of a mathend000#, that is H = a$\mbox{${\mathbb Z}$}$ mathend000#.

Theorem 4   Let G mathend000# be a multiplicative group with neutral element e mathend000#. Let x $\in$ G mathend000# an element and gr(x mathend000#) the subgroup of H mathend000# consisting of all powers of x mathend000# (including e = x⁰ mathend000# and x^-1 mathend000# the inverse of x mathend000#). Let $\Theta$(x) mathend000# be the order of gr(x mathend000#), that is the cardinality of gr(x mathend000#). Then two cases arise

• either gr(x mathend000#) is infinite and then the powers of x mathend000# are pairwise different and thus H mathend000# is isomorphic to $\mbox{${\mathbb Z}$}$ mathend000#.
• or gr(x mathend000#) is finite and we have the following properties
  1. $\Theta$(x) mathend000# is the smallest integer n mathend000# such that xⁿ = e mathend000#,
  2. x^m = x^m' mathend000# iff m $\equiv$ m'mod $\Theta$(x) mathend000#,
  3. H mathend000# is isomorphic to ($\mbox{${\mathbb Z}$}$/n$\mbox{${\mathbb Z}$}$) $\setminus$ {0} mathend000# where n = $\Theta$(x) mathend000#,
  4. H = {e, x, x²,..., x^n-1} mathend000# where n = $\Theta$(x) mathend000#.

Theorem 5 (Lagrange)   For every subgroup H mathend000# of the finite group G mathend000#, the order (that is the cardinality) of H mathend000# divides that of G mathend000#.