Groups: Subgroups of S_3

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  • Опубликовано: 23 дек 2024

Комментарии • 19

  • @lemyul
    @lemyul 5 лет назад +19

    so an S7 is going to be a pain in the ass

  • @quibblequack4350
    @quibblequack4350 Год назад

    Thank you, it's crystal clear!

  • @parkeredmisten6639
    @parkeredmisten6639 Год назад

    Why is (123)^2 = (132)?

    • @AdamGlesser
      @AdamGlesser  Год назад

      Think of an equilateral triangle where you label the corners 1, 2, 3 starting from one vertex and moving counterclockwise. The permutation (1 2 3) will rotate the triangle 120 degrees counterclockwise sending corner 1 to position 2, corner 2 to position 3 and corner 3 to position 1. If you perform this same move again (the equivalent of permuting by (1 2 3)² in total), then corner 1 will go to position 3, corner 2 will go to position 1, and corner 3 will go to position 2. This is the same as if you had rotated 120 clockwise from the beginning, which is given by the permutation (1 3 2).
      Alternatively, you can just write down (1 2 3) ⚬ (1 2 3) and compute its value on each number:
      [(1 2 3) ⚬ (1 2 3)](1) = (1 2 3)(2) = 3
      [(1 2 3) ⚬ (1 2 3)](2) = (1 2 3)(3) = 1
      [(1 2 3) ⚬ (1 2 3)](3) = (1 2 3)(1) = 2
      and we see that (1 2 3) ⚬ (1 2 3) = (1 3 2).

  • @abidkakar403
    @abidkakar403 2 года назад

    Sir
    Write the elements of length of the group of S3
    Please answer

    • @AdamGlesser
      @AdamGlesser  2 года назад

      I don't understand the question. Would you clarify what it is you want to know?

  • @maiquynhnghinguyen9414
    @maiquynhnghinguyen9414 2 года назад

    can you explain why (1 2)(1 3) =( 1 3 2) ?

    • @AdamGlesser
      @AdamGlesser  2 года назад

      The permutation on the left should be read in two pieces, from right to left. The first piece says to swap 1 and 3, the second to swap 1 and 2. Think of the numbers as representing positions of objects. So when you swap 1 and 3 it means to swap the objects in positions 1 and 3. So where does the object starting in position 1 go. The (1 3) moves it to position 3. The (1 2) then leaves it alone. In total, the object moved from position 1 to position 3. Now, where does the object in position 3 go. The (1 3) moves it to position 1 and then the (1 2) moves it to position 2. In total, in moved from position 3 to position 2. Finally, the object in position 2 is left alone by (1 3) and then moved to position 1 by (1 2). Thus, the object in position 2 is moved to position 1. We summarize all of this with (1 3 2).
      Also, consider watching my video on composing permutations: ruclips.net/video/ii5NXuoH148/видео.html

  • @gaaraofddarkness
    @gaaraofddarkness 3 года назад

    Thanks Adam

  • @shoopinc
    @shoopinc 5 лет назад +1

    Thank you

  • @hamzashah925
    @hamzashah925 3 года назад

    Write subgroup of s4 plz

  • @The1Gujjar
    @The1Gujjar 5 лет назад +1

    Thanks 😘

  • @mongky9903
    @mongky9903 3 года назад

    Why H6 not be just { e, (1 2 3) }

    • @AdamGlesser
      @AdamGlesser  3 года назад +1

      The set you listed is not closed. The square of (1 2 3) needs be contained in the subgroup. But the square equals (1 3 2), which is not in the set you gave.

    • @mongky9903
      @mongky9903 3 года назад

      @@AdamGlesser Tks u very much sir, now i see that you already mention it on the video. Great work btw!

    • @AdamGlesser
      @AdamGlesser  3 года назад +1

      @@mongky9903 Glad to help!

  • @tariqmath1845
    @tariqmath1845 2 месяца назад

    Thank you