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

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

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

Marc Moreno Maza
2008-01-07