Computing orbits of a $S_5$ group action

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Consider the group action of $S_5$ on $(mathbbZ/5mathbbZ)^6$ given by $$(12345)colon (a,b,c,d,e,f)mapsto(b,c,d,e,a,f)$$
$$(12)colon (a,b,c,d,e,f)mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$
Do we know how to compute the number of orbits of the action? Do we know how to calculate the length of each orbit? I am interested in what possible stabilizer subgroup groups would appear.

To me it appears this problem is this is too large to calculate by hand, and I hope someone could help to solve it by computer programming.

abstract-algebra symmetric-groups group-actions computational-algebra

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edited Jul 21 at 7:29

Mike Pierce

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asked Jul 20 at 23:34

Qixiao

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Consider the group action of $S_5$ on $(mathbbZ/5mathbbZ)^6$ given by $$(12345)colon (a,b,c,d,e,f)mapsto(b,c,d,e,a,f)$$
$$(12)colon (a,b,c,d,e,f)mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$
Do we know how to compute the number of orbits of the action? Do we know how to calculate the length of each orbit? I am interested in what possible stabilizer subgroup groups would appear.

To me it appears this problem is this is too large to calculate by hand, and I hope someone could help to solve it by computer programming.

abstract-algebra symmetric-groups group-actions computational-algebra

share|cite|improve this question

edited Jul 21 at 7:29

Mike Pierce

11k93574

asked Jul 20 at 23:34

Qixiao

2,6891525

add a comment | 

up vote
3
down vote

favorite

up vote
3
down vote

favorite

Consider the group action of $S_5$ on $(mathbbZ/5mathbbZ)^6$ given by $$(12345)colon (a,b,c,d,e,f)mapsto(b,c,d,e,a,f)$$
$$(12)colon (a,b,c,d,e,f)mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$
Do we know how to compute the number of orbits of the action? Do we know how to calculate the length of each orbit? I am interested in what possible stabilizer subgroup groups would appear.

To me it appears this problem is this is too large to calculate by hand, and I hope someone could help to solve it by computer programming.

abstract-algebra symmetric-groups group-actions computational-algebra

share|cite|improve this question

edited Jul 21 at 7:29

Mike Pierce

11k93574

asked Jul 20 at 23:34

Qixiao

2,6891525

Consider the group action of $S_5$ on $(mathbbZ/5mathbbZ)^6$ given by $$(12345)colon (a,b,c,d,e,f)mapsto(b,c,d,e,a,f)$$
$$(12)colon (a,b,c,d,e,f)mapsto(-a,-b,f-b-e,d,f-c-a,f-b-a)$$
Do we know how to compute the number of orbits of the action? Do we know how to calculate the length of each orbit? I am interested in what possible stabilizer subgroup groups would appear.

To me it appears this problem is this is too large to calculate by hand, and I hope someone could help to solve it by computer programming.

abstract-algebra symmetric-groups group-actions computational-algebra

share|cite|improve this question

edited Jul 21 at 7:29

Mike Pierce

11k93574

asked Jul 20 at 23:34

Qixiao

2,6891525

share|cite|improve this question

share|cite|improve this question

edited Jul 21 at 7:29

Mike Pierce

11k93574

edited Jul 21 at 7:29

Mike Pierce

11k93574

edited Jul 21 at 7:29

Mike Pierce

11k93574

11k93574

asked Jul 20 at 23:34

Qixiao

2,6891525

asked Jul 20 at 23:34

Qixiao

2,6891525

asked Jul 20 at 23:34

Qixiao

2,6891525

2,6891525

add a comment | 

add a comment | 

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For computing the number of orbits I suggest Burnside's lemma:
https://en.wikipedia.org/wiki/Burnside%27s_lemma

It is a very good tradeoff. Instead of doing calculations over the $5^6$ elements of the underlying set, you just have to compute the number of fixed points of 120 permutations, and of course, many cases are similar.

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answered Jul 21 at 0:06

A. Pongrácz

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
3
down vote

For computing the number of orbits I suggest Burnside's lemma:
https://en.wikipedia.org/wiki/Burnside%27s_lemma

It is a very good tradeoff. Instead of doing calculations over the $5^6$ elements of the underlying set, you just have to compute the number of fixed points of 120 permutations, and of course, many cases are similar.

share|cite|improve this answer

answered Jul 21 at 0:06

A. Pongrácz

2,309221

add a comment | 

up vote
3
down vote

For computing the number of orbits I suggest Burnside's lemma:
https://en.wikipedia.org/wiki/Burnside%27s_lemma

It is a very good tradeoff. Instead of doing calculations over the $5^6$ elements of the underlying set, you just have to compute the number of fixed points of 120 permutations, and of course, many cases are similar.

share|cite|improve this answer

answered Jul 21 at 0:06

A. Pongrácz

2,309221

add a comment | 

up vote
3
down vote

up vote
3
down vote

For computing the number of orbits I suggest Burnside's lemma:
https://en.wikipedia.org/wiki/Burnside%27s_lemma

It is a very good tradeoff. Instead of doing calculations over the $5^6$ elements of the underlying set, you just have to compute the number of fixed points of 120 permutations, and of course, many cases are similar.

share|cite|improve this answer

answered Jul 21 at 0:06

A. Pongrácz

2,309221

For computing the number of orbits I suggest Burnside's lemma:
https://en.wikipedia.org/wiki/Burnside%27s_lemma

It is a very good tradeoff. Instead of doing calculations over the $5^6$ elements of the underlying set, you just have to compute the number of fixed points of 120 permutations, and of course, many cases are similar.

share|cite|improve this answer

answered Jul 21 at 0:06

A. Pongrácz

2,309221

share|cite|improve this answer

share|cite|improve this answer

answered Jul 21 at 0:06

A. Pongrácz

2,309221

answered Jul 21 at 0:06

A. Pongrácz

2,309221

answered Jul 21 at 0:06

A. Pongrácz

2,309221

2,309221

add a comment | 

add a comment | 

 

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