Why write permutations as disjoint cycles and transpositions?

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Apologies in advance if this isn’t an appropriate question for this website. I was reading an introduction to groups book which explained you can write permutations as a product of disjoint cycles and then further that you can write a cycle as a product of transpositions. What is the benefit of thinking about a permutation this way? I do kind of see that by grouping the numbers into the cycles, you can see which numbers are involved with each other and which numbers are unaffected by the rearrangement the permutation is describing.

abstract-algebra permutations permutation-cycles

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edited Jul 23 at 1:08

Bill Wallis

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asked Jul 14 at 22:08

anonanon444

1396

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions.
  – thesmallprint
  Jul 14 at 22:21
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_n$ is the set of all permutations on $n$ elements, and that $A_n$ is the set of all even permutations on $n$ elements. The group $A_n$ is of great importance in classifying some groups of particular orders, for example.
  – Bill Wallis
  Jul 14 at 22:24
  
  

  
  
  
  


• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot!
  – dbx
  Jul 14 at 22:30
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions.
  – Michal Adamaszek
  Jul 14 at 22:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar.
  – anonanon444
  Jul 14 at 23:44
  

  
  
  
  

 | 
show 1 more comment

up vote
4
down vote

favorite

2

Apologies in advance if this isn’t an appropriate question for this website. I was reading an introduction to groups book which explained you can write permutations as a product of disjoint cycles and then further that you can write a cycle as a product of transpositions. What is the benefit of thinking about a permutation this way? I do kind of see that by grouping the numbers into the cycles, you can see which numbers are involved with each other and which numbers are unaffected by the rearrangement the permutation is describing.

abstract-algebra permutations permutation-cycles

share|cite|improve this question

edited Jul 23 at 1:08

Bill Wallis

1,8271822

asked Jul 14 at 22:08

anonanon444

1396

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions.
  – thesmallprint
  Jul 14 at 22:21
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_n$ is the set of all permutations on $n$ elements, and that $A_n$ is the set of all even permutations on $n$ elements. The group $A_n$ is of great importance in classifying some groups of particular orders, for example.
  – Bill Wallis
  Jul 14 at 22:24
  
  

  
  
  
  


• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot!
  – dbx
  Jul 14 at 22:30
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions.
  – Michal Adamaszek
  Jul 14 at 22:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar.
  – anonanon444
  Jul 14 at 23:44
  

  
  
  
  

 | 
show 1 more comment

up vote
4
down vote

favorite

2

up vote
4
down vote

favorite

2

2

Apologies in advance if this isn’t an appropriate question for this website. I was reading an introduction to groups book which explained you can write permutations as a product of disjoint cycles and then further that you can write a cycle as a product of transpositions. What is the benefit of thinking about a permutation this way? I do kind of see that by grouping the numbers into the cycles, you can see which numbers are involved with each other and which numbers are unaffected by the rearrangement the permutation is describing.

abstract-algebra permutations permutation-cycles

share|cite|improve this question

edited Jul 23 at 1:08

Bill Wallis

1,8271822

asked Jul 14 at 22:08

anonanon444

1396

Apologies in advance if this isn’t an appropriate question for this website. I was reading an introduction to groups book which explained you can write permutations as a product of disjoint cycles and then further that you can write a cycle as a product of transpositions. What is the benefit of thinking about a permutation this way? I do kind of see that by grouping the numbers into the cycles, you can see which numbers are involved with each other and which numbers are unaffected by the rearrangement the permutation is describing.

abstract-algebra permutations permutation-cycles

share|cite|improve this question

edited Jul 23 at 1:08

Bill Wallis

1,8271822

asked Jul 14 at 22:08

anonanon444

1396

share|cite|improve this question

share|cite|improve this question

edited Jul 23 at 1:08

Bill Wallis

1,8271822

edited Jul 23 at 1:08

Bill Wallis

1,8271822

edited Jul 23 at 1:08

Bill Wallis

1,8271822

1,8271822

asked Jul 14 at 22:08

anonanon444

1396

asked Jul 14 at 22:08

anonanon444

1396

asked Jul 14 at 22:08

anonanon444

1396

1396

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions.
  – thesmallprint
  Jul 14 at 22:21
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_n$ is the set of all permutations on $n$ elements, and that $A_n$ is the set of all even permutations on $n$ elements. The group $A_n$ is of great importance in classifying some groups of particular orders, for example.
  – Bill Wallis
  Jul 14 at 22:24
  
  

  
  
  
  


• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot!
  – dbx
  Jul 14 at 22:30
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions.
  – Michal Adamaszek
  Jul 14 at 22:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar.
  – anonanon444
  Jul 14 at 23:44
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions.
  – thesmallprint
  Jul 14 at 22:21
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_n$ is the set of all permutations on $n$ elements, and that $A_n$ is the set of all even permutations on $n$ elements. The group $A_n$ is of great importance in classifying some groups of particular orders, for example.
  – Bill Wallis
  Jul 14 at 22:24
  
  

  
  
  
  


• 
  
  
  

  
  6
  

  
  
  
  
  
  

  
  It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot!
  – dbx
  Jul 14 at 22:30
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions.
  – Michal Adamaszek
  Jul 14 at 22:44
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar.
  – anonanon444
  Jul 14 at 23:44
  

  
  
  
  

1

1

as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions.
– thesmallprint
Jul 14 at 22:21

as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions.
– thesmallprint
Jul 14 at 22:21

3

3

Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_n$ is the set of all permutations on $n$ elements, and that $A_n$ is the set of all even permutations on $n$ elements. The group $A_n$ is of great importance in classifying some groups of particular orders, for example.
– Bill Wallis
Jul 14 at 22:24

Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_n$ is the set of all permutations on $n$ elements, and that $A_n$ is the set of all even permutations on $n$ elements. The group $A_n$ is of great importance in classifying some groups of particular orders, for example.
– Bill Wallis
Jul 14 at 22:24

6

6

It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot!
– dbx
Jul 14 at 22:30

It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot!
– dbx
Jul 14 at 22:30

1

1

This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions.
– Michal Adamaszek
Jul 14 at 22:44

This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions.
– Michal Adamaszek
Jul 14 at 22:44

1

1

I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar.
– anonanon444
Jul 14 at 23:44

I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar.
– anonanon444
Jul 14 at 23:44

 | 
show 1 more comment

1 Answer
1

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2
down vote



accepted

Factoring into disjoint cycles (a transposition is a cycle) is usually the most "useful" way of looking at a permutation. Two main reasons (though there are probably lots of others):

1. It makes it very clear what the orbits of a permutation are.


2. It makes it very easy to compute the order of the permutation (LCM of the cycle lengths).

share|cite|improve this answer

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

add a comment | 

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1 Answer
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1 Answer
1

active

oldest

votes

active

oldest

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active

oldest

votes

up vote
2
down vote



accepted

Factoring into disjoint cycles (a transposition is a cycle) is usually the most "useful" way of looking at a permutation. Two main reasons (though there are probably lots of others):

1. It makes it very clear what the orbits of a permutation are.


2. It makes it very easy to compute the order of the permutation (LCM of the cycle lengths).

share|cite|improve this answer

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

add a comment | 

up vote
2
down vote



accepted

Factoring into disjoint cycles (a transposition is a cycle) is usually the most "useful" way of looking at a permutation. Two main reasons (though there are probably lots of others):

1. It makes it very clear what the orbits of a permutation are.


2. It makes it very easy to compute the order of the permutation (LCM of the cycle lengths).

share|cite|improve this answer

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

add a comment | 

up vote
2
down vote



accepted

up vote
2
down vote



accepted

Factoring into disjoint cycles (a transposition is a cycle) is usually the most "useful" way of looking at a permutation. Two main reasons (though there are probably lots of others):

1. It makes it very clear what the orbits of a permutation are.


2. It makes it very easy to compute the order of the permutation (LCM of the cycle lengths).

share|cite|improve this answer

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

Factoring into disjoint cycles (a transposition is a cycle) is usually the most "useful" way of looking at a permutation. Two main reasons (though there are probably lots of others):

1. It makes it very clear what the orbits of a permutation are.


2. It makes it very easy to compute the order of the permutation (LCM of the cycle lengths).

share|cite|improve this answer

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

share|cite|improve this answer

share|cite|improve this answer

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

answered Jul 15 at 0:56

Morgan Rodgers

9,07421338

9,07421338

add a comment | 

add a comment | 

 

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