Generators and Cyclic Subgroups

This chapter provides introduction on generating subgroups from elements in finite Abelian groups; definitions and examples of subgroup generators, subgroup order, cyclic subgroups, Lagrange-Theorem.

What Is Subgroup in Abelian Group

What Is Subgroup Generator in Abelian Group

What Is Order of Element

Every Element Is Subgroup Generator

Order of Subgroup and Lagrange Theorem

What Is Cyclic Group

Element Generated Subgroup Is Cyclic

Takeaways:

• A subgroup in an Abelian Group is a subset of the Abelian Group that itself is an Abelian Group.
• A subgroup generator is an element in an finite Abelian Group that can be used to generate a subgroup using a series of scalar multiplication operations in additive notation.
• The order of an element P in a finite Abelian Group is the least positive integer n, such that the scalar multiplication of n and P is 0, where 0 is the identity element in additive notation.
• Every element in a finite Abelian group is a subgroup generator, which can be used to generate an subgroup.
• Lagrange Theorem states that the order of any subgroup in an finite Abelian group divides the order of the parent group.
• A Cyclic Subgroup is a finite Abelian group that can be generated by a single element using the scalar multiplication operation in additive notation (or exponentiation operation in multiplicative notation).
• All subgroups generated from elements in finite Abelian groups are cyclic groups.