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  • 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.

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循環群(抽象代數) (Cyclic Groups (Abstract Algebra))

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    林宜悉 發佈於 2021 年 01 月 14 日
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