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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 {an|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 aG such thata〉, 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〉 = {an|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, a2, a3, a4,…,an−1,an = e

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