We say a group G is a cyclic group if it can be generated by a single element.
To understand this definition and notation, we must first explain what it means
for a group to be generated by an element.
Once we’ve done that, we’ll give several examples,
explain why the word “cyclic” was chosen for this definition,
and then finally talk about why these types of groups are so important.
When working with groups, you typically use additive notation or multiplicative notation.
This is done even if the elements of the group are not numbers and the group operation
is not numerical, but is instead something like geometric transformations or function composition.
When using additive notation, the identity element is denoted by 0,
and when using multiplicative notation, the identity element is denoted by 1.
But keep thinking abstractly,
even if the notation tries to lure your mind into the familiar realm of the real numbers…
Let’s now dive into the definition of cyclic groups.
Let G be any group, and pick an element 'x' in G.
Here’s a puzzle: what’s the smallest subgroup of G that contains 'x'?
First, any subgroup that contains 'x' must also contain its inverse…
It also has to contain the identity element…
And to be closed under the group operation, it has to contain all powers of 'x'...
and all powers of the inverse of 'x'...
This set of all integral powers of 'x' is the smallest subgroup of G containing 'x'.
We call it the group generated by 'x' and denote it using brackets.
If G contains an element 'x' such that G equals the group generated by 'x',
then we say G is a cyclic group.
It’s worth taking a moment to repeat this definition using additive notation.
Let H be a group, and pick an element 'y' in H.
The group generated by 'y' is the smallest subgroup of H containing 'y'.
It must contain 'y', its inverse '-y', and the identity element 0.
And to be a group it must contain all positive and negative multiples of 'y'.
If H can be generated by an element 'y', then we say H is a cyclic group.
Let’s look at a few examples of cyclic groups.
A classic example is the group of integers under addition.
The integers are generated by the number 1.
To see this, remember the group generated by 1 must contain:
1, the identity element 0, the additive inverse of 1 (which is -1),
and it must also contain all multiples of 1 and -1.
This covers all the integers.
The integers are a cyclic group!
The integers are an example of an infinite cyclic group.
Let’s now look at a FINITE cyclic group.
The classic example is the integers mod N under addition.
This is a finite group with N elements.
It is also generated by the number 1.
But something different happens here.
Look at all the positive and negative multiples of 1.
Recall that 'n' is congruent to 0 mod 'n'…
n + 1 is congruent to 1 Mod 'n', and so on.
-1 is congruent to N-1, -2 is congruent to N-2, and so on..
So the group generated by 1 repeats itself.
It cycles through the numbers 0 through N-1 over and over.
This is why it’s called a “cyclic group.”
The integers mod N are a finite, cyclic group under addition.
In abstract algebra, the integers mod N are written like this.
This will make sense once you’ve studied quotient groups,
so don’t panic if you're not familiar with this notation.
We’ve now seen two types of cyclic groups: the integers Z under addition, which is infinite,
and the integers mod N under addition, which is finite.
Are there other cyclic groups?
No! This is it!
The complete collection of cyclic groups.
The integers.
The integers mod 2.
The integers mod 3…
The integers mod 4, and so forth.
Oh, and don’t forget the trivial group.
Why are cyclic groups so important?
The big reason is due to a result known as
The Fundamental Theorem of Finitely Generated Abelian Groups
That’s quite a title!
What it says is that any abelian group that is finitely generated can be broken apart
into a finite number of cyclic groups.
And every cyclic group is either the integers, or the integers mod N.
So cyclic groups are the fundamental building blocks for finitely generated abelian groups.
It takes a lot of work to understand and prove this theorem,
but you’ve just taken your first step…