# Tour:Equivalence of definitions of cyclic group

**This article adapts material from the main article:** equivalence of definitions of cyclic group

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Cyclic group|UP: Introduction four (beginners)|NEXT: Every nontrivial subgroup of the group of integers is cyclic on its smallest element

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WHAT YOU NEED TO DO:

- Read, and thoroughly understand, the equivalence of the two definitions of cyclic group. (We'll see better formulations of the proof later, after introducing homomorphisms and quotients)

## 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.

## 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.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Cyclic group|UP: Introduction four (beginners)|NEXT: Every nontrivial subgroup of the group of integers is cyclic on its smallest element