Some results from group theory

Definition 4   Let be a (multiplicative) group with neutral element . A nonempty subset is a subgroup of if the following three statements hold
• ,
• for every we have ,
• for every the inverse of belongs to .

Theorem 3   For every subgroup of the additive abelian group there exists an element such that is the set of the multiples of , that is .

Theorem 4   Let be a multiplicative group with neutral element . Let an element and gr( ) the subgroup of consisting of all powers of (including and the inverse of ). Let be the order of gr( ), that is the cardinality of gr( ). Then two cases arise
• either gr( ) is infinite and then the powers of are pairwise different and thus is isomorphic to .
• or gr( ) is finite and we have the following properties
1. is the smallest integer such that ,
2. iff ,
3. is isomorphic to where ,
4. where .

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

Marc Moreno Maza
2008-01-07