Writing a Permutation as a product of Disjoint CyclesHow to Write Permutation as the Product of Transpositions?Writing a Permutation as a product of Disjoint CyclesHow to Write Permutation as the Product of Transpositions?Permutation of cycles as transpositionExpressing a permuation as a product of disjoint cycles.Abstract product of disjoint cyclesPermutation written as product of transpositionsIf a permutation is a cycle, can I factorize it as a product of disjoint cycles?Permutation and Disjoint cycles questionProve that every permutation is the product of disjoint cyclesPermutation as disjoint cycles

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Writing a Permutation as a product of Disjoint Cycles


How to Write Permutation as the Product of Transpositions?Writing a Permutation as a product of Disjoint CyclesHow to Write Permutation as the Product of Transpositions?Permutation of cycles as transpositionExpressing a permuation as a product of disjoint cycles.Abstract product of disjoint cyclesPermutation written as product of transpositionsIf a permutation is a cycle, can I factorize it as a product of disjoint cycles?Permutation and Disjoint cycles questionProve that every permutation is the product of disjoint cyclesPermutation as disjoint cycles













1












$begingroup$


How do i write a permutation as a product of disjoint cycles ?



enter image description here



I know that in order to determine a cycle we need to start with the smallest element and move on till the mapping points to itself.Then start with the next non repeating smallest element..But how to write this as a product of disjoint cycles?










share|cite|improve this question











$endgroup$











  • $begingroup$
    I suppose it is worth mentioning that disjoint cycles commute, that is, may be multiplied in any order. Hence (4)(1532) is also a valid answer.
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:19










  • $begingroup$
    @A.Sh From the method i have described i got (1532)(4).Is this itself the way to get Disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:21










  • $begingroup$
    Well, yes, you do (always) end up with disjoint cycles by performing your algorithm. Also, if you do exactly as described, you will end up with (1532)(4), so you have done correctly. Since disjoint cycles are permutations that act on entirely different sets of elements, they can be applied in any order. Just mentioning it, in case you would encounter a similar situation in the future : )
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:31










  • $begingroup$
    @A.Sh okay.thanks
    $endgroup$
    – techno
    Oct 8 '15 at 9:38















1












$begingroup$


How do i write a permutation as a product of disjoint cycles ?



enter image description here



I know that in order to determine a cycle we need to start with the smallest element and move on till the mapping points to itself.Then start with the next non repeating smallest element..But how to write this as a product of disjoint cycles?










share|cite|improve this question











$endgroup$











  • $begingroup$
    I suppose it is worth mentioning that disjoint cycles commute, that is, may be multiplied in any order. Hence (4)(1532) is also a valid answer.
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:19










  • $begingroup$
    @A.Sh From the method i have described i got (1532)(4).Is this itself the way to get Disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:21










  • $begingroup$
    Well, yes, you do (always) end up with disjoint cycles by performing your algorithm. Also, if you do exactly as described, you will end up with (1532)(4), so you have done correctly. Since disjoint cycles are permutations that act on entirely different sets of elements, they can be applied in any order. Just mentioning it, in case you would encounter a similar situation in the future : )
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:31










  • $begingroup$
    @A.Sh okay.thanks
    $endgroup$
    – techno
    Oct 8 '15 at 9:38













1












1








1


0



$begingroup$


How do i write a permutation as a product of disjoint cycles ?



enter image description here



I know that in order to determine a cycle we need to start with the smallest element and move on till the mapping points to itself.Then start with the next non repeating smallest element..But how to write this as a product of disjoint cycles?










share|cite|improve this question











$endgroup$




How do i write a permutation as a product of disjoint cycles ?



enter image description here



I know that in order to determine a cycle we need to start with the smallest element and move on till the mapping points to itself.Then start with the next non repeating smallest element..But how to write this as a product of disjoint cycles?







group-theory permutations






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Oct 8 '15 at 11:46









Andrés E. Caicedo

65.7k8160251




65.7k8160251










asked Oct 8 '15 at 7:07









technotechno

3222826




3222826











  • $begingroup$
    I suppose it is worth mentioning that disjoint cycles commute, that is, may be multiplied in any order. Hence (4)(1532) is also a valid answer.
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:19










  • $begingroup$
    @A.Sh From the method i have described i got (1532)(4).Is this itself the way to get Disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:21










  • $begingroup$
    Well, yes, you do (always) end up with disjoint cycles by performing your algorithm. Also, if you do exactly as described, you will end up with (1532)(4), so you have done correctly. Since disjoint cycles are permutations that act on entirely different sets of elements, they can be applied in any order. Just mentioning it, in case you would encounter a similar situation in the future : )
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:31










  • $begingroup$
    @A.Sh okay.thanks
    $endgroup$
    – techno
    Oct 8 '15 at 9:38
















  • $begingroup$
    I suppose it is worth mentioning that disjoint cycles commute, that is, may be multiplied in any order. Hence (4)(1532) is also a valid answer.
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:19










  • $begingroup$
    @A.Sh From the method i have described i got (1532)(4).Is this itself the way to get Disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:21










  • $begingroup$
    Well, yes, you do (always) end up with disjoint cycles by performing your algorithm. Also, if you do exactly as described, you will end up with (1532)(4), so you have done correctly. Since disjoint cycles are permutations that act on entirely different sets of elements, they can be applied in any order. Just mentioning it, in case you would encounter a similar situation in the future : )
    $endgroup$
    – A.Sh
    Oct 8 '15 at 7:31










  • $begingroup$
    @A.Sh okay.thanks
    $endgroup$
    – techno
    Oct 8 '15 at 9:38















$begingroup$
I suppose it is worth mentioning that disjoint cycles commute, that is, may be multiplied in any order. Hence (4)(1532) is also a valid answer.
$endgroup$
– A.Sh
Oct 8 '15 at 7:19




$begingroup$
I suppose it is worth mentioning that disjoint cycles commute, that is, may be multiplied in any order. Hence (4)(1532) is also a valid answer.
$endgroup$
– A.Sh
Oct 8 '15 at 7:19












$begingroup$
@A.Sh From the method i have described i got (1532)(4).Is this itself the way to get Disjoint cycles?
$endgroup$
– techno
Oct 8 '15 at 7:21




$begingroup$
@A.Sh From the method i have described i got (1532)(4).Is this itself the way to get Disjoint cycles?
$endgroup$
– techno
Oct 8 '15 at 7:21












$begingroup$
Well, yes, you do (always) end up with disjoint cycles by performing your algorithm. Also, if you do exactly as described, you will end up with (1532)(4), so you have done correctly. Since disjoint cycles are permutations that act on entirely different sets of elements, they can be applied in any order. Just mentioning it, in case you would encounter a similar situation in the future : )
$endgroup$
– A.Sh
Oct 8 '15 at 7:31




$begingroup$
Well, yes, you do (always) end up with disjoint cycles by performing your algorithm. Also, if you do exactly as described, you will end up with (1532)(4), so you have done correctly. Since disjoint cycles are permutations that act on entirely different sets of elements, they can be applied in any order. Just mentioning it, in case you would encounter a similar situation in the future : )
$endgroup$
– A.Sh
Oct 8 '15 at 7:31












$begingroup$
@A.Sh okay.thanks
$endgroup$
– techno
Oct 8 '15 at 9:38




$begingroup$
@A.Sh okay.thanks
$endgroup$
– techno
Oct 8 '15 at 9:38










3 Answers
3






active

oldest

votes


















5












$begingroup$

First, we note that writing it as a product of disjoint cycles means that each number appears only once throughout all of the cycles.



We see that $1mapsto 5$, $5mapsto 3$, $3mapsto 2$, $2mapsto 1$. So, we can express this in cycle notation as
$$(1532).$$
Now, we see what is left over... well, that is just $4$, which is fixed by the permutation in question. So, the permutation can be written as



$$(1532)(4),mbox or equivalently, just (1532).$$






share|cite|improve this answer











$endgroup$








  • 1




    $begingroup$
    Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:20










  • $begingroup$
    That's all you need to do :)
    $endgroup$
    – eloiprime
    Oct 8 '15 at 7:20










  • $begingroup$
    okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
    $endgroup$
    – techno
    Oct 8 '15 at 7:22






  • 1




    $begingroup$
    Yes, I mean, that's all I did above. Wasn't it?
    $endgroup$
    – eloiprime
    Oct 8 '15 at 8:00










  • $begingroup$
    Yeah...got it now
    $endgroup$
    – techno
    Oct 8 '15 at 9:39


















0












$begingroup$

$4$ is the only one invariant, while the others mingle among them, thus there are only two cycles
$$(1,5,3,2)$$
It's customary not to write any singleton.






share|cite|improve this answer









$endgroup$




















    0












    $begingroup$

    You start with the first number than you go to the number with is under this number and so on until you get back to the start number...... In your case you start with 1 then go to 5 then to 3 to 2 and are back at one,.... So the first cycle is (1;5;2;3)



    Then you look at the numbers which aren't in the first cycle e.g. 4 and do the same again.... ( you can, but you do not have to mention cycles with length 1 if it is known in which Sn the permutation is or it is otherwise clear how long the permutation is.)






    share|cite|improve this answer









    $endgroup$












    • $begingroup$
      okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
      $endgroup$
      – techno
      Oct 8 '15 at 7:23











    Your Answer





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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    First, we note that writing it as a product of disjoint cycles means that each number appears only once throughout all of the cycles.



    We see that $1mapsto 5$, $5mapsto 3$, $3mapsto 2$, $2mapsto 1$. So, we can express this in cycle notation as
    $$(1532).$$
    Now, we see what is left over... well, that is just $4$, which is fixed by the permutation in question. So, the permutation can be written as



    $$(1532)(4),mbox or equivalently, just (1532).$$






    share|cite|improve this answer











    $endgroup$








    • 1




      $begingroup$
      Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
      $endgroup$
      – techno
      Oct 8 '15 at 7:20










    • $begingroup$
      That's all you need to do :)
      $endgroup$
      – eloiprime
      Oct 8 '15 at 7:20










    • $begingroup$
      okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
      $endgroup$
      – techno
      Oct 8 '15 at 7:22






    • 1




      $begingroup$
      Yes, I mean, that's all I did above. Wasn't it?
      $endgroup$
      – eloiprime
      Oct 8 '15 at 8:00










    • $begingroup$
      Yeah...got it now
      $endgroup$
      – techno
      Oct 8 '15 at 9:39















    5












    $begingroup$

    First, we note that writing it as a product of disjoint cycles means that each number appears only once throughout all of the cycles.



    We see that $1mapsto 5$, $5mapsto 3$, $3mapsto 2$, $2mapsto 1$. So, we can express this in cycle notation as
    $$(1532).$$
    Now, we see what is left over... well, that is just $4$, which is fixed by the permutation in question. So, the permutation can be written as



    $$(1532)(4),mbox or equivalently, just (1532).$$






    share|cite|improve this answer











    $endgroup$








    • 1




      $begingroup$
      Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
      $endgroup$
      – techno
      Oct 8 '15 at 7:20










    • $begingroup$
      That's all you need to do :)
      $endgroup$
      – eloiprime
      Oct 8 '15 at 7:20










    • $begingroup$
      okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
      $endgroup$
      – techno
      Oct 8 '15 at 7:22






    • 1




      $begingroup$
      Yes, I mean, that's all I did above. Wasn't it?
      $endgroup$
      – eloiprime
      Oct 8 '15 at 8:00










    • $begingroup$
      Yeah...got it now
      $endgroup$
      – techno
      Oct 8 '15 at 9:39













    5












    5








    5





    $begingroup$

    First, we note that writing it as a product of disjoint cycles means that each number appears only once throughout all of the cycles.



    We see that $1mapsto 5$, $5mapsto 3$, $3mapsto 2$, $2mapsto 1$. So, we can express this in cycle notation as
    $$(1532).$$
    Now, we see what is left over... well, that is just $4$, which is fixed by the permutation in question. So, the permutation can be written as



    $$(1532)(4),mbox or equivalently, just (1532).$$






    share|cite|improve this answer











    $endgroup$



    First, we note that writing it as a product of disjoint cycles means that each number appears only once throughout all of the cycles.



    We see that $1mapsto 5$, $5mapsto 3$, $3mapsto 2$, $2mapsto 1$. So, we can express this in cycle notation as
    $$(1532).$$
    Now, we see what is left over... well, that is just $4$, which is fixed by the permutation in question. So, the permutation can be written as



    $$(1532)(4),mbox or equivalently, just (1532).$$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Oct 5 '18 at 8:06

























    answered Oct 8 '15 at 7:14









    eloiprimeeloiprime

    2,1651721




    2,1651721







    • 1




      $begingroup$
      Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
      $endgroup$
      – techno
      Oct 8 '15 at 7:20










    • $begingroup$
      That's all you need to do :)
      $endgroup$
      – eloiprime
      Oct 8 '15 at 7:20










    • $begingroup$
      okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
      $endgroup$
      – techno
      Oct 8 '15 at 7:22






    • 1




      $begingroup$
      Yes, I mean, that's all I did above. Wasn't it?
      $endgroup$
      – eloiprime
      Oct 8 '15 at 8:00










    • $begingroup$
      Yeah...got it now
      $endgroup$
      – techno
      Oct 8 '15 at 9:39












    • 1




      $begingroup$
      Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
      $endgroup$
      – techno
      Oct 8 '15 at 7:20










    • $begingroup$
      That's all you need to do :)
      $endgroup$
      – eloiprime
      Oct 8 '15 at 7:20










    • $begingroup$
      okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
      $endgroup$
      – techno
      Oct 8 '15 at 7:22






    • 1




      $begingroup$
      Yes, I mean, that's all I did above. Wasn't it?
      $endgroup$
      – eloiprime
      Oct 8 '15 at 8:00










    • $begingroup$
      Yeah...got it now
      $endgroup$
      – techno
      Oct 8 '15 at 9:39







    1




    1




    $begingroup$
    Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:20




    $begingroup$
    Thanks.As i have described in the question what i get by applying is (1532)(4) =>(1532).Is this itself the method to get disjoint cycles?
    $endgroup$
    – techno
    Oct 8 '15 at 7:20












    $begingroup$
    That's all you need to do :)
    $endgroup$
    – eloiprime
    Oct 8 '15 at 7:20




    $begingroup$
    That's all you need to do :)
    $endgroup$
    – eloiprime
    Oct 8 '15 at 7:20












    $begingroup$
    okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
    $endgroup$
    – techno
    Oct 8 '15 at 7:22




    $begingroup$
    okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles.
    $endgroup$
    – techno
    Oct 8 '15 at 7:22




    1




    1




    $begingroup$
    Yes, I mean, that's all I did above. Wasn't it?
    $endgroup$
    – eloiprime
    Oct 8 '15 at 8:00




    $begingroup$
    Yes, I mean, that's all I did above. Wasn't it?
    $endgroup$
    – eloiprime
    Oct 8 '15 at 8:00












    $begingroup$
    Yeah...got it now
    $endgroup$
    – techno
    Oct 8 '15 at 9:39




    $begingroup$
    Yeah...got it now
    $endgroup$
    – techno
    Oct 8 '15 at 9:39











    0












    $begingroup$

    $4$ is the only one invariant, while the others mingle among them, thus there are only two cycles
    $$(1,5,3,2)$$
    It's customary not to write any singleton.






    share|cite|improve this answer









    $endgroup$

















      0












      $begingroup$

      $4$ is the only one invariant, while the others mingle among them, thus there are only two cycles
      $$(1,5,3,2)$$
      It's customary not to write any singleton.






      share|cite|improve this answer









      $endgroup$















        0












        0








        0





        $begingroup$

        $4$ is the only one invariant, while the others mingle among them, thus there are only two cycles
        $$(1,5,3,2)$$
        It's customary not to write any singleton.






        share|cite|improve this answer









        $endgroup$



        $4$ is the only one invariant, while the others mingle among them, thus there are only two cycles
        $$(1,5,3,2)$$
        It's customary not to write any singleton.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Oct 8 '15 at 7:15









        MASLMASL

        708313




        708313





















            0












            $begingroup$

            You start with the first number than you go to the number with is under this number and so on until you get back to the start number...... In your case you start with 1 then go to 5 then to 3 to 2 and are back at one,.... So the first cycle is (1;5;2;3)



            Then you look at the numbers which aren't in the first cycle e.g. 4 and do the same again.... ( you can, but you do not have to mention cycles with length 1 if it is known in which Sn the permutation is or it is otherwise clear how long the permutation is.)






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
              $endgroup$
              – techno
              Oct 8 '15 at 7:23
















            0












            $begingroup$

            You start with the first number than you go to the number with is under this number and so on until you get back to the start number...... In your case you start with 1 then go to 5 then to 3 to 2 and are back at one,.... So the first cycle is (1;5;2;3)



            Then you look at the numbers which aren't in the first cycle e.g. 4 and do the same again.... ( you can, but you do not have to mention cycles with length 1 if it is known in which Sn the permutation is or it is otherwise clear how long the permutation is.)






            share|cite|improve this answer









            $endgroup$












            • $begingroup$
              okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
              $endgroup$
              – techno
              Oct 8 '15 at 7:23














            0












            0








            0





            $begingroup$

            You start with the first number than you go to the number with is under this number and so on until you get back to the start number...... In your case you start with 1 then go to 5 then to 3 to 2 and are back at one,.... So the first cycle is (1;5;2;3)



            Then you look at the numbers which aren't in the first cycle e.g. 4 and do the same again.... ( you can, but you do not have to mention cycles with length 1 if it is known in which Sn the permutation is or it is otherwise clear how long the permutation is.)






            share|cite|improve this answer









            $endgroup$



            You start with the first number than you go to the number with is under this number and so on until you get back to the start number...... In your case you start with 1 then go to 5 then to 3 to 2 and are back at one,.... So the first cycle is (1;5;2;3)



            Then you look at the numbers which aren't in the first cycle e.g. 4 and do the same again.... ( you can, but you do not have to mention cycles with length 1 if it is known in which Sn the permutation is or it is otherwise clear how long the permutation is.)







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Oct 8 '15 at 7:19









            BörgeBörge

            1,037415




            1,037415











            • $begingroup$
              okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
              $endgroup$
              – techno
              Oct 8 '15 at 7:23

















            • $begingroup$
              okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
              $endgroup$
              – techno
              Oct 8 '15 at 7:23
















            $begingroup$
            okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
            $endgroup$
            – techno
            Oct 8 '15 at 7:23





            $begingroup$
            okay.. so if i use the method i described in the question ie: finding the smallest etc... i will get the product of disjoint cycles?
            $endgroup$
            – techno
            Oct 8 '15 at 7:23


















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