A cyclic group is a mathematical object that is formed by a set of elements and a binary operation that combines two elements to produce a third element. The elements in a cyclic group are generated by repeatedly applying the binary operation to a starting element, known as the generator.
A group is considered cyclic if there exists a single element, known as the generator, that can generate all the other elements in the group by repeatedly applying the group operation. In other words, if every element in the group can be expressed as a power of the generator, then the group is cyclic.
One of the main properties of a cyclic group is that it is abelian, meaning that the group operation is commutative. Additionally, every cyclic group is finite and has a finite number of elements. Lastly, every subgroup of a cyclic group is also cyclic.
No, a non-abelian group cannot be cyclic. This is because the defining property of a cyclic group is that it is abelian, and a non-abelian group, by definition, does not satisfy this property.
Cyclic groups play a crucial role in many areas of mathematics, including number theory, abstract algebra, and group theory. They also have applications in cryptography and coding theory. Additionally, cyclic groups are used to study other mathematical objects, such as fields and vector spaces.