Study of cyclic groups form a crucial part in group theory. Here are some important definitions, facts and examples of cyclic groups.

#### Cyclic group and its generator

Let be a group and . Then the set forms a subgroup of G and is called the subgroup generated by .

If for some , then G is called a **cyclic group** and is referred to as it’s **generator**.

#### Examples

- Integer modulo n,
- whenever
- , for

#### Counterexamples

- , where p is any prime.
- Any non-abelian group.

#### Important facts

- Any cyclic group is abelian.
- Any cyclic group is either finite or countable.
- Any subgroup of cyclic group is cyclic.
- Any quotient group of cyclic group is cyclic.
- Any group of prime order is cyclic.
- Any group G, with , , where p and q are distinct primes is cyclic.
- Any non-abelian group is non-cyclic.
- Any group which is uncountable is not cyclic.
- If is a generator of G, then is also a generator of G.
- Any infinite cyclic group has exactly two generators and is isomorphic to .
- Any finite cyclic group G is isomorphic to , where .

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