Tour:Equivalence of definitions of cyclic group
This article adapts material from the main article: equivalence of definitions of cyclic group
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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.
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