## Subtitles section Play video

• 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 weve done that, well give several examples,

• explain why the wordcyclicwas 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 youve studied quotient groups,

• so don’t panic if you're not familiar with this notation.

• Weve 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 youve just taken your first step

We say a group G is a cyclic group if it can be generated by a single element.

Subtitles and vocabulary

Click the word to look it up Click the word to find further inforamtion about it

# Cyclic Groups (Abstract Algebra)

• 72 5
林宜悉 posted on 2020/03/06
Video vocabulary