# Equivalence of definitions of cyclic group

This article gives a proof/explanation of the equivalence of multiple definitions for the term cyclic group

View a complete list of pages giving proofs of equivalence of definitions

## Contents

## The definitions that we have to prove as equivalent

### Definition in terms of modular arithmetic

A group is said to be **cyclic** (sometimes, **monogenic** or **monogenous**) if it is either isomorphic to the group of integers or to the group of integers modulo n for some positive integer .

### Definition in terms of generating sets

A group is termed **cyclic** (sometimes, **monogenic** or **monogenous**) if it has a generating set of size 1.

### Definition as a quotient

A group is termed **cyclic** if it is a quotient of the group , in other words, there exists a surjective homomorphism from to the group.

## Proof of equivalence

### From modular arithmetic to generating sets

This is direct: is generated by the element , and is generated by the element 1.

### From generating sets to modular arithmetic

**Given**: A group with a generating set

**To prove**: is isomorphic either to (the group of integers) or to (the group of integers modulo n)

**Proof**: We consider two cases.

**Case 1**: has finite order. Thus, there exists a minimal positive integer such that is the identity element. Consider now the map that sends to the element . We want to prove that is an isomorphism.

We first show that . For this, observe that if and add up to less than as integers, then by definition. If the sum of and as integers is at least , then (since is the identity element).

Similarly, by definition, and , again because .

*Surjectivity*: Since generates , every element of can be written as a power of , say for some integer . Writing where are integers and , we get that . Thus, is surjective.

*Injectivity*: Finally, if with both in , then , contradicting the assumption that has order .

Thus, is an isomorphism of groups.

**Case 2**: does not have finite order. In that case, consider the map that sends to .

Clearly, by definition, , , and .

*Surjectivity*: Since generates , every element of can be written as for some integer .

*Injectivity*: If for , then , contradicting the assumption that does not have finite order.