## Chapter 8

## Cyclic Groups

In the previous chapter we have defined a cyclic subgroup of a group. Recall that if *a* is an element of a group *G,* then {*a*^{n}|*n* ∈ ℤ} is a subgroup of G, called the cyclic subgroup of G generated by *a* and is written as 〈*a*〉. In this chapter we shall study cyclic groups and their properties.

#### 8.1 Definition and Examples

**Definition 8.1.** *A group G is said to be cyclic if there exists some a* ∈ *G such that* 〈*a*〉, *the subgroup generated by a is whole of G. The element a is called a generator of G or G is said to be generated by a.*

*Thus G* = 〈*a*〉 = {*a*^{n}|*n* ∈ ℤ}. *If the binary operation is addition, then G* = 〈*a*〉 = {*na*|*n* ∈ ℤ}.

**Remark 8.1.** *If G is a finite cyclic group of order n, generated by a, then G* = {*a, a*^{2}, *a*^{3}, *a*^{4},…,*a*^{n−1},*a*^{n} = *e*