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.