a list of the conjugacy classes of the symetric groups S1 through S6 are as foll

Question

a list of the conjugacy classes of the symetric groups S1 through S6 are as follows...

S1 is 1

S2 is 2

S3 is 4

S4 is 11

S5 is 19

S6 is 56

Now I suspect that these are also the number of subgroups (non isomorphic) including the group itself of the respective symetric group....I basically need this conjecture confirmed by an expert...If you want you can elaborate how the conjugacy classes reveal the number of different subgroups...

I have done a search algorithm on Mathematica for the purposes of music that gave me numbers for supposedly all subgroups of the symetric groups S1 through S6 which doubtlessly contain duplicates when isomorphism is considered but "sound" different due to the different permutation elements in each isomorphic subgroup...

the numbers I have found are as follows

S1 is 1

S2 is 2

S3 is 6

S4 is 30

S5 is 156

S6 is 1310

I do not know if this is an exhaustive list though...to generate these subgroups I chose every combination of 2 from the entire element list of the respective group and composed till closure was attained then I took the Union of my results to eliminate the nonisomorphic copies....

any feed back would be appreciated...dont spend to much time on your answer....I dont want to be "flagged"

Explanation / Answer

I am not sure this is allowed, but i am going to try this once to see what happens.... I am going to answer my own question...the "Online encyclopedia of Integer sequences" confirmed that the numbers of the conjugacy classes are also the number of different subgroups of the respective symetric group when isomorphism is taken into consideration... Further the second list of numbers is correct for what I claimed they represented except that S6 is incorrect....

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.