Order of Elements in a Group | Abstract Algebra

We introduce the order of group elements in this Abstract Algebra lessons. We'll see the definition of the order of an element in a group, several examples of finding the order of an element in a group, and we will introduce two basic but important results concerning distinct powers of elements with finite order and elements with infinite order. #abstractalgebra #grouptheory

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[WrathofMath]

I previously had a video on this, but I used a needlessly complicated definition which made the lesson a few minutes longer than it needed to be. So I redid it.

[keldonchase]

Thanks for the videos; they are immensely helpful!!!
Question about the example presented at 4:40:
The way I look at it is:

Cycle 1:
Start with 1:
1 —-> 6 (1)
6 —-> 4 (2)
4 —> 2 (3)
2 —> 1 (4)
I started with 1 and ended with 1 so to get back to 1, I had to make 4 “jumps” so 4 is a possible answer for our order.

Cycle 2:
Skip all the numbers covered in Cycle 1 so I start with 3.
3 ——> 3 (0)
Since 3 maps to itself, I do not consider a “jump”.
0 would be a possible answer but since it is non-positive, I can eliminate it as a possibility.

Cycle 3:
5 ——> 5 (0)
Again, not a “jump” because 5 maps to itself.
Again, 0 can be eliminated as a possibility since it is non-positive.

So that is why the order is 4.
Is this way of thinking correct or is there something important I’m missing?
Thank you!!

[Dravignor]

If e is the identity element of G, and e is also a power of some a ∈ G, then can we say ord(e) = 0, or is it ord(e) = 1 because the order of an element is strictly for non-zero positive integers?

Edit: Nevermind I watched the next video

[mgyodzs]

Good explanation! Keep going, You do good job.

[MrCoreyTexas]

I was wondering why not 0 for the order, and the reason it is not 0 is because every element to the 0th power is the identity element, so it has to be a positive integer to be non trivial

[punditgi]

You can never go wrong with Wrath of Math!

[InoceramusGigas]

Hi W.O.M

Would love some videos on ramsey theory!

Could be a great fit in your graph theory playlist, or just within a general combinatorics theme.

There is definitely room for a more intuitive explanation on YouTube.

Love the Abstract algebra vids.

[nyxiiestelle]

am i the only one who couldn't understand this 😭😭? I'm feeling very dumb, i watched it so many times and still can't get it well idk why🥹!! ( I'm a math major)