The proof of properties of dihedral group and its commutative elements

Abstract

In the field of abstract algebra, there is an important branch for the foundations of mathematics, namely group. The group is a non-empty set with some certain axioms. In the group, there is an interesting part, that is finite group. For the finite group has been defined the commutativity degree as the comparasion between the number of commutative elements of group and its order. If the finite group is commutative, then its commutativity degree is one. If the finite group is not commutative, then its commutativity degree is less than one. One example of a non-commutative finite group is a dihedral group. A dihedral group is a group which elements are the result of a composition of two permutations with predetermined properties. Keith Conrad in his article entitled “dihedral group” specifically discussed about the special properties of the group. Keith Conrad defined dihedral groups as a result of reflection and rotation operations. All the properties of dihedral groups are proven by geometry approach as the reflection and rotation. The purpose of this study is to prove the properties of dihedral group using functions, furthermore this research will examine which elements are commutative in dihedral group.