cyclic group (idea) by -brazil-

A cyclic group is a mathematical group which is generated by one of its elements, i.e. every element x can be written as x = a^k, where a is the generator and k is an integer.

Cyclic groups are important in number theory because any cyclic group of infinite order is isomorphic to the group formed by the set of all integers and addition as the operation, and any finite cyclic group of order n is isomorphic to the set of all integers modulo n, using addition followed by taking the modulo as the operation.

Fundamental theorem of finite abelian groups	group theory	proof that the multiplicative group of a finite field is cyclic	Lagrange's Theorem
Shor's Algorithm	Generator	nth roots of unity	abelian group
proof of Lagrange's theorem for groups	classification theorem of finite simple groups	number theory	group
Lagrange, Indiana	Division Algorithm	Well Ordering Theorem	integers modulo n
How to solve linear equations with matrices	Euler Phi function	ElGamal	finite field
relatively prime	greatest common divisor	Subgroup	cyclic