字幕表 動画を再生する
-
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…