Cyclic Groups  (Abstract Algebra)

Cyclic Groups (Abstract Algebra)


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…

100 thoughts to “Cyclic Groups (Abstract Algebra)”

  1. Everything was good. I have my exams tomorrow and I found this good piece. I'm so happy.
    And oh, that music at the end…lovely!

  2. Amazing channel. Very very Thanks Socratica it is because of you I am able to obtain good marks in Group theory….

  3. Why group should contain all multiples of 1 under operation addition.? Isnt (1,0,-1) a sufficient group under addition?

  4. Thank you Socratica! Your videos are wonderful and have given me new determination to continue my abstract algebra course. One of your greatest strengths (for me personally), is having well explained examples to accompany all your definitions and explanations. Most lecturers at the Tertiary level just tend to spout off theorems and definitions without actually showing any applicability to what we are learning- so this has completely changed the way I understand my maths. I will definitely be sharing this with all of my peers. Thank you again 🙂

  5. mam can u please cover whole theorem of abstract algebra like suppose in cyclic group please cover all the theorem.I hope u will make video .Thanku

  6. What do you mean by "complete collection" of cyclic groups at 3:44 ? There can be more cyclic groups defined on other operations right?

  7. This is incredibly powerful teaching! Like this 95% of Americans could study and graduate in university Mathematics, my deep respect!

  8. I have yet to take an actual class on abstract algebra (been studying it on my own because I find it fun), but these videos really fill in the holes of my understanding. I knew a fascination with axiomatic set theory would be useful someday!

  9. thanks socrtica,,thanks to you I don't hate abstarct algebra any more,,,wish you all the best,,,keep on doing such incredible and understanding videos,,,

  10. Cube roots of unity also form a cyclic group but u mentioned in the video that other than those two there are no cyclic groups.pls verify it.If I am wrong pls notify me.

  11. That is a great idea to have a beautiful face and voice person to teach a boring subject! That makes sure viewer not easy to fall asleep and keeping one awake.

  12. The diagram for Integers mod n under addition has shown me the light. Thank you gorgeous stranger with the voice of power and mind of algebra.

  13. Hi, excellent work and explanation. You say that all the cyclic groups are the integers or the integers mod n. I guess you were trying to avoid the use of isomorphic (correct me if I am wrong ) to avoid confusion. Maybe using “behaves as , or is identical to “ instead , but my idea could end up confusing people even more 😊 but there I let my 3cents. Critical students will be able to create groups that are not the integers and are cyclical.

  14. Wow, first video in this series I've watched and it seems like a much more productive use of my time than the MIT or similar series of lectures for abstract algebra.

  15. Hey thanks!! That's was great help for me to learn basics of abstract algebra.I would say this is the series everyone watch after 3blue1brown's essence of linear algebra.

  16. It’s a perfect explanation that I wants… well done..Keep it up.we need more from you..
    May you get 1million subscribers soon..

  17. Thank you for providing an easy way to understand this stuff. Honestly this stuff isn't hard – you could probably teach it in high school actually. But everything online reads like you already need a BS in math or something.

  18. Ok I just deduced the notion of prime element using abstract algebra and IT REALLY EXISTS
    OMG I was like playing around with theorems and this came out
    Math is so intuitive:)))

  19. Math is an Art, and abstract algebra is poetry. Treat Math as only a Science, and you will relegate it to the utilitarian. Treat it as an Art, and you will begin to converse in a language that transcends common communication.

  20. Me gusta mucho tus videos. No tienes una version en español?

    I like so much your videos do you have a spanish vertion?

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