Name: 環状群（抽象代数 (Cyclic Groups (Abstract Algebra))
Uploaded: 2021-01-14T08:49:33.000Z
Duration: 5 min 1 s
Description: VoiceTubeの動画で発音を聞きながら英語表現を覚えよう！学べる英語：

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.

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'...

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',

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.

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.

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.

The complete collection of cyclic groups.

Oh, and don’t forget the trivial group.

The big reason is due to a result known as

The Fundamental Theorem of Finitely Generated Abelian Groups

What it says is that any abelian group that is finitely generated can be broken apart

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…