GT1. Definition of Group

Abstract Algebra: We introduce the notion of a group and describe basic properties. Examples given include familiar abelian groups and the symmetric groups.

[docnelson]

Thank you MathDoctorBob-I am a retired physics Phd studying advanced math for fun. Your videos are superb and much appreciated.

[gremlinextreme]

Got a bit behind in my abstract class, I'm going to watch all 49 videos in a row. Sunday fun day

[christinebrun]

Please return MathDoctorBob, I love your videos!

[MathDoctorBob]

There is a rule for orders of permutations (lcm of lengths of disjoint cycles). Of course you need to experiment to get a feel for the important examples. For (12)(345), just take powers until you get to the identity.

(12)(345) square (354) cube (12) fourth (345) fifth (12)(354) sixth e

[MathDoctorBob]

Once we note binary operation in the definition, closure is redundant. Most of the time, we verify closure instead of binary operation though. It's also worth restating as a condition because: 1) it's usually necessary to check in examples, 2) subgroup = group inside of a group, and 3) in a less strenuous environment (ie. talking to high school students about math), we may want to bypass mentioning binary operations.

[tondekush]

This helps so much all the way in Zimbabwe. Thank you Doctor Bob

[MathDoctorBob]

The were named by Galois, although Lagrange worked with them earlier. We use groups to encode symmetry, so they appear everywhere in the real world.

Some important groups in physics are rotation groups and translation groups. Einstein's theory of special relativity is based on the Poincare group. In a different vein, finite groups are at the heart of the discrete and fast Fourier transforms, which occur all over signal processing.

[mewhoi]

Thank you Dr. Bob, I found the problem sets you mentioned! And I am in the modern algebra class at school, watching your lectures help me a lot! Thank you!!!
Best,
Sarah

[vinteachesmath]

Thank you, this video was excellent. I plan on teaching myself Modern Algebra this summer. I will definitely supplement with this playlist!

[kirkgio]

im so glad that there is video like this...so much understand..im filipino studting mathematics...

[ecksor]

I second the Fraleigh recommendation, that book helped me a lot when I struggled with the course text for a Galois Theory class.

[jmich]

What a fantastic teacher you are Bob, thank you so much indeed!

[MathDoctorBob]

I use Herstein's Topics in Algebra as an outline, but he doesn't do group actions. A good book for beginners is Fraleigh's A First Course in Algebra. Some people will recommend Artin, or Dummit and Foote; Save them for grad school.

[ElvinJones]

Around 8:50 you start using cycle notation. I am having difficulty following this. Do you do a video that explains this notation more slowly and with a bit more detail?

[MathDoctorBob]

I've come across that. It's amazing what you can find on the web now.

[MathDoctorBob]

You're welcome! The most horrifying thing I've seen in Abstract Algebra is presenting rings and fields before groups. It makes sense historically, but ay! Different teaching goals, if any.

[PDucksChannel]

Thank You SOOO Much, I'm 15 and a sophomore in high school, and thanks to you and a lot of other videos and books I've been able to teach myself the one thing I love more than anything...math! Sadly I have to be in a Algebra 2 class because no one will help me get moved up. :( Seriously though thank you, for everything!

[MathDoctorBob]

@weweallthewayhome In the works. I've held off on this for two reasons: I need a playlist on div/grad/curl and baby Stoke's, etc. for background, and I have a set of Gene Calabi's notes on curves and surfaces in storage somewhere.

If you need any immediate help, let me know. My game is Lie groups and vector bundles. - Bob

[MathDoctorBob]

Your welcome and good luck! If you have any questions, I'm happy to help.

[MathDoctorBob]

You're welcome! Glad to be of help.