What's the difference between permutations and derangements? Are the latter applicable to the group theory universe, or are they more relevant to (an)other branch(es) of math(s)?
@@shruggzdastr8-facedclown Permutations are arrangements of elements where the order matters, and every element is assigned a distinct position. A derangement is a specific kind of permutation where no element appears in its original position. Permutations are very important in group theory (because of what is shown in this video). Derangements often appear in combinatorics, especially in problems related to probability and counting. In group theory, derangements are not very useful but could still arise in certain contexts, like studying specific subgroups of the symmetric group. I can’t think of any specific example right now though
@dibeos : I asked because I saw a video on either or both Numberphile and/or Stand-Up Maths (due to the involvement of Matt Parker -- who appears frequently on the former channel and who owns the latter one) on derangements a few/several years ago, and your description of permutations reminded me of what I remembered being discussed in that video. Thanks for answering so quickly as well as for clarifying that derangements aren't typically relevant to group theory (accept for certain highly-specialized situations)
@dibeos : Have you ever done a video on John Conway's work in Game Theory (also his Game Of Life, which developed from that work -- but, maybe that should be saved for another video) or Gödel's Incompleteness Theorem?
@@shruggzdastr8-facedclown we talked about the incompleteness theorem in one of our videos, but a whole video dedicated to it where we go in depth would be a good idea… 🤔
@@dibeos I'm confused since it is a long time since I've done groups but thanks for showing me. It's just that you wrote p3(3)=3*4=2 when I thought it should be p3(4)=3*4=2
@MW-dg7gl it’s ok no problem. You are doing right by asking, that’s the way we learn and “relearn” things, plus sometimes Sofia and I make mistakes so thanks for trying to point one out anyway
@dibeos I don't mind the microphones. I didn't watch for the microphones. I watched for the visualizers and information. The mics had 0 affect on my enjoyment of this video. Was not jarring for me, neither. Ideas were conveyed, but there is obv an expectation of prior knowledge of groups and permutations. I mentioned that because I did find it peculiar that you chose to define wbat 1 to 1 is, and a few other things, but not isomorphic. Small criticism really. I likes the video nonetheless .
@@Arycke thanks for the honest criticism, really. I think one of the hardest things in educational content is know how to measure the level of of depth in each subject. Now that you mentioned it really seems disproportionate. Thanks again, we are getting better!!!
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And permutations are just matrices, which are also used to study group properties through matrix multiplication. I liked the video.
It's Yoneda Lemma in the case G is considered as a category with a single object whose set of endomorphisms is the group itself ❤️
Can you make a video about alternating groups? As in Sn->An and how they are used algorithms? Its been a subject of my interest
@@DanHorus899 absolutely! 😎
It's interesting that An is normal in Sn and simple. So you can only talk about permutation parity, not permutation modulo anything higher.
What's the difference between permutations and derangements? Are the latter applicable to the group theory universe, or are they more relevant to (an)other branch(es) of math(s)?
@@shruggzdastr8-facedclown Permutations are arrangements of elements where the order matters, and every element is assigned a distinct position. A derangement is a specific kind of permutation where no element appears in its original position. Permutations are very important in group theory (because of what is shown in this video). Derangements often appear in combinatorics, especially in problems related to probability and counting. In group theory, derangements are not very useful but could still arise in certain contexts, like studying specific subgroups of the symmetric group. I can’t think of any specific example right now though
@dibeos : I asked because I saw a video on either or both Numberphile and/or Stand-Up Maths (due to the involvement of Matt Parker -- who appears frequently on the former channel and who owns the latter one) on derangements a few/several years ago, and your description of permutations reminded me of what I remembered being discussed in that video. Thanks for answering so quickly as well as for clarifying that derangements aren't typically relevant to group theory (accept for certain highly-specialized situations)
@shruggzdastr8-facedclown exactly! You’re welcome 😉 please let us know what kind of content would you like to see on the channel 😎
@dibeos : Have you ever done a video on John Conway's work in Game Theory (also his Game Of Life, which developed from that work -- but, maybe that should be saved for another video) or Gödel's Incompleteness Theorem?
@@shruggzdastr8-facedclown we talked about the incompleteness theorem in one of our videos, but a whole video dedicated to it where we go in depth would be a good idea… 🤔
Excellent video! Are you two a couple? ❤
@@naakatube yes, we are married 😎
8:31 typo?
@@MW-dg7gl I don’t see any typo. What exactly doesn’t look right? 🤔
@@dibeos p3(4)=3*4=2?, my guess from looking at the pattern.
nope…that’s indeed the case, take a look at the table at 6:53
@@dibeos I'm confused since it is a long time since I've done groups but thanks for showing me. It's just that you wrote p3(3)=3*4=2 when I thought it should be p3(4)=3*4=2
@MW-dg7gl it’s ok no problem. You are doing right by asking, that’s the way we learn and “relearn” things, plus sometimes Sofia and I make mistakes so thanks for trying to point one out anyway
I don't think this approach works. I found it jarring. You also need to find a better solution than the huge microphones, which look ridiculous.
Thanks for the comment. Please, let us know how to get better.
@dibeos I don't mind the microphones. I didn't watch for the microphones. I watched for the visualizers and information. The mics had 0 affect on my enjoyment of this video. Was not jarring for me, neither. Ideas were conveyed, but there is obv an expectation of prior knowledge of groups and permutations.
I mentioned that because I did find it peculiar that you chose to define wbat 1 to 1 is, and a few other things, but not isomorphic. Small criticism really. I likes the video nonetheless .
@@Arycke thanks for the honest criticism, really. I think one of the hardest things in educational content is know how to measure the level of of depth in each subject. Now that you mentioned it really seems disproportionate. Thanks again, we are getting better!!!