Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?












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Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?






abstract-algebra permutations notation







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    $begingroup$
    There is no difference.
    $endgroup$
    – David G. Stork
    Dec 24 '18 at 20:14










  • $begingroup$
    @Shaun Thanks I did it
    $endgroup$
    – BrianH
    Dec 24 '18 at 20:48
















0












$begingroup$


Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?






abstract-algebra permutations notation







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$endgroup$








  • 8




    $begingroup$
    There is no difference.
    $endgroup$
    – David G. Stork
    Dec 24 '18 at 20:14










  • $begingroup$
    @Shaun Thanks I did it
    $endgroup$
    – BrianH
    Dec 24 '18 at 20:48














0












0








0





$begingroup$


Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?






abstract-algebra permutations notation







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$endgroup$




Is there a difference between $(2,,4) circ (1,,3)$ and $(2,,4)(1,,3)$?






abstract-algebra permutations notation




abstract-algebra permutations notation






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edited Dec 24 '18 at 20:19









Shaun

11.1k113688




11.1k113688










asked Dec 24 '18 at 20:13









BrianHBrianH

658




658








  • 8




    $begingroup$
    There is no difference.
    $endgroup$
    – David G. Stork
    Dec 24 '18 at 20:14










  • $begingroup$
    @Shaun Thanks I did it
    $endgroup$
    – BrianH
    Dec 24 '18 at 20:48














  • 8




    $begingroup$
    There is no difference.
    $endgroup$
    – David G. Stork
    Dec 24 '18 at 20:14










  • $begingroup$
    @Shaun Thanks I did it
    $endgroup$
    – BrianH
    Dec 24 '18 at 20:48








8




8




$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14




$begingroup$
There is no difference.
$endgroup$
– David G. Stork
Dec 24 '18 at 20:14












$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48




$begingroup$
@Shaun Thanks I did it
$endgroup$
– BrianH
Dec 24 '18 at 20:48










3 Answers
3






active

oldest

votes


















0












$begingroup$

Here $(24)(13)$ refers to the permutation
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix},
end{align}
in two-line notation,
whereas $(24)$ and $(13)$ are the permutations
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
1 & 4 & 3 & 2
end{pmatrix}

text{ and }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4
end{pmatrix}
end{align}
respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
begin{align}
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 2 & 1 & 4\
3 & 4 & 1 & 2
end{pmatrix} text{which simplifies to }
begin{pmatrix}
1 & 2 & 3 & 4\
3 & 4 & 1 & 2
end{pmatrix}.
end{align}
Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.






share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).






    share|cite|improve this answer









    $endgroup$





















      1












      $begingroup$

      $(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.






      share|cite|improve this answer









      $endgroup$









      • 1




        $begingroup$
        Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
        $endgroup$
        – Eike Schulte
        Dec 24 '18 at 22:22










      • $begingroup$
        Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
        $endgroup$
        – Shaun
        Dec 27 '18 at 13:36












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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      0












      $begingroup$

      Here $(24)(13)$ refers to the permutation
      begin{align}
      begin{pmatrix}
      1 & 2 & 3 & 4\
      3 & 4 & 1 & 2
      end{pmatrix},
      end{align}
      in two-line notation,
      whereas $(24)$ and $(13)$ are the permutations
      begin{align}
      begin{pmatrix}
      1 & 2 & 3 & 4\
      1 & 4 & 3 & 2
      end{pmatrix}

      text{ and }
      begin{pmatrix}
      1 & 2 & 3 & 4\
      3 & 2 & 1 & 4
      end{pmatrix}
      end{align}
      respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
      begin{align}
      begin{pmatrix}
      1 & 2 & 3 & 4\
      3 & 2 & 1 & 4\
      3 & 4 & 1 & 2
      end{pmatrix} text{which simplifies to }
      begin{pmatrix}
      1 & 2 & 3 & 4\
      3 & 4 & 1 & 2
      end{pmatrix}.
      end{align}
      Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        Here $(24)(13)$ refers to the permutation
        begin{align}
        begin{pmatrix}
        1 & 2 & 3 & 4\
        3 & 4 & 1 & 2
        end{pmatrix},
        end{align}
        in two-line notation,
        whereas $(24)$ and $(13)$ are the permutations
        begin{align}
        begin{pmatrix}
        1 & 2 & 3 & 4\
        1 & 4 & 3 & 2
        end{pmatrix}

        text{ and }
        begin{pmatrix}
        1 & 2 & 3 & 4\
        3 & 2 & 1 & 4
        end{pmatrix}
        end{align}
        respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
        begin{align}
        begin{pmatrix}
        1 & 2 & 3 & 4\
        3 & 2 & 1 & 4\
        3 & 4 & 1 & 2
        end{pmatrix} text{which simplifies to }
        begin{pmatrix}
        1 & 2 & 3 & 4\
        3 & 4 & 1 & 2
        end{pmatrix}.
        end{align}
        Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          Here $(24)(13)$ refers to the permutation
          begin{align}
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 4 & 1 & 2
          end{pmatrix},
          end{align}
          in two-line notation,
          whereas $(24)$ and $(13)$ are the permutations
          begin{align}
          begin{pmatrix}
          1 & 2 & 3 & 4\
          1 & 4 & 3 & 2
          end{pmatrix}

          text{ and }
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 2 & 1 & 4
          end{pmatrix}
          end{align}
          respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
          begin{align}
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 2 & 1 & 4\
          3 & 4 & 1 & 2
          end{pmatrix} text{which simplifies to }
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 4 & 1 & 2
          end{pmatrix}.
          end{align}
          Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.






          share|cite|improve this answer









          $endgroup$



          Here $(24)(13)$ refers to the permutation
          begin{align}
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 4 & 1 & 2
          end{pmatrix},
          end{align}
          in two-line notation,
          whereas $(24)$ and $(13)$ are the permutations
          begin{align}
          begin{pmatrix}
          1 & 2 & 3 & 4\
          1 & 4 & 3 & 2
          end{pmatrix}

          text{ and }
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 2 & 1 & 4
          end{pmatrix}
          end{align}
          respectively. When we compose $(24)$ with $(13)$ to get $(24)circ (13)$ we mean
          begin{align}
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 2 & 1 & 4\
          3 & 4 & 1 & 2
          end{pmatrix} text{which simplifies to }
          begin{pmatrix}
          1 & 2 & 3 & 4\
          3 & 4 & 1 & 2
          end{pmatrix}.
          end{align}
          Hence the composition has the same two-line expression as $(24)(13)$ so $(24)circ(13) = (24)(13)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 24 '18 at 20:29









          Jacky ChongJacky Chong

          19.3k21129




          19.3k21129























              2












              $begingroup$

              The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).






              share|cite|improve this answer









              $endgroup$


















                2












                $begingroup$

                The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).






                share|cite|improve this answer









                $endgroup$
















                  2












                  2








                  2





                  $begingroup$

                  The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).






                  share|cite|improve this answer









                  $endgroup$



                  The first one is the result of $(2, 4)$ on the left side and $(1, 3)$ on the right side of the binary operation $circ$, commonly understood to mean the composition of the two permutations as functions (unless stated otherwise); the second is the concatenation of $(2, 4)$ and $(1, 3)$ and is commonly understood to mean the same thing (when the context is clear).







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 24 '18 at 20:23









                  ShaunShaun

                  11.1k113688




                  11.1k113688























                      1












                      $begingroup$

                      $(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.






                      share|cite|improve this answer









                      $endgroup$









                      • 1




                        $begingroup$
                        Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
                        $endgroup$
                        – Eike Schulte
                        Dec 24 '18 at 22:22










                      • $begingroup$
                        Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
                        $endgroup$
                        – Shaun
                        Dec 27 '18 at 13:36
















                      1












                      $begingroup$

                      $(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.






                      share|cite|improve this answer









                      $endgroup$









                      • 1




                        $begingroup$
                        Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
                        $endgroup$
                        – Eike Schulte
                        Dec 24 '18 at 22:22










                      • $begingroup$
                        Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
                        $endgroup$
                        – Shaun
                        Dec 27 '18 at 13:36














                      1












                      1








                      1





                      $begingroup$

                      $(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.






                      share|cite|improve this answer









                      $endgroup$



                      $(2,,4)(1,,3)$ is a misuse of the multiplication record instead of composing. We have the same example in matrix multiplication. In fact, it is not about multiplication of matrices, but about composing a matrix. So $A B$ is actualy a matrix composition, not a matrix multiplication.







                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Dec 24 '18 at 20:21









                      Maria MazurMaria Mazur

                      50.7k1362126




                      50.7k1362126








                      • 1




                        $begingroup$
                        Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
                        $endgroup$
                        – Eike Schulte
                        Dec 24 '18 at 22:22










                      • $begingroup$
                        Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
                        $endgroup$
                        – Shaun
                        Dec 27 '18 at 13:36














                      • 1




                        $begingroup$
                        Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
                        $endgroup$
                        – Eike Schulte
                        Dec 24 '18 at 22:22










                      • $begingroup$
                        Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
                        $endgroup$
                        – Shaun
                        Dec 27 '18 at 13:36








                      1




                      1




                      $begingroup$
                      Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
                      $endgroup$
                      – Eike Schulte
                      Dec 24 '18 at 22:22




                      $begingroup$
                      Calling every use of concatenation (i.e. “multiplication notation”) for anything other than multiplication a misuse is probably not helpful. The notation is too useful and there are many things similar enough to multiplication that restricting yourself to multiplication of numbers is simply infeasible. It’s better to get used to it because others will stop using it.
                      $endgroup$
                      – Eike Schulte
                      Dec 24 '18 at 22:22












                      $begingroup$
                      Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
                      $endgroup$
                      – Shaun
                      Dec 27 '18 at 13:36




                      $begingroup$
                      Adding to this, most of semigroup theory is done using concatenation as the binary operation. It'd be unsightly and impractical otherwise.
                      $endgroup$
                      – Shaun
                      Dec 27 '18 at 13:36


















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