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
The definitions that we have to prove as equivalent
Definition in terms of modular arithmetic
Definition in terms of generating sets
Definition as a quotient
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
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.