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GROUPS:
Group Theory

Right Cosets

Bruce Colletti
Posted 10 years ago

For G a subgroup of S(n) -- the symmetric group on n-letters -- does the code below return all right cosets of G in S(n)?

cosets = GroupOrbits[G, GroupElements@Sn, PermutationProduct[#1, #2] &];

If so, is there another way to obtain the cosets?

The code seemed to work until I hit an example for n = 7, at which time Mathematica disputed GAP ... and that ain't good, which means I may not correctly understand how to use the third argument in GroupOrbits.

Thankx.
POSTED BY: Bruce Colletti
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Yes, Bruce. That's correct, and a good way to express it.

Jose.
POSTED BY: Jose Martin-Garcia
Bruce Colletti
Bruce Colletti
Posted 10 years ago

Thanks for the reply Jose. Let's make sure I understand.

Let G be a group and X be a pointlist.

The GroupOrbits documentation addresses GroupOrbits[G, X, f[X-element, G-element]].

Thus X-elements and G-elements are referenced by #1 and #2 respectively -- their positions as f-arguments.

Since left cosets of G have form x G={x g: g in G}, each term x g is given by PermutationProduct[#1, #2].

Since right cosets of G have form G x={g x: g in G}, each term g x is given by PermutationProduct[#2, #1].

Did I get this right? If so, I'll tell Tech Support to ignore the notebook sent Sunday evening. After it was sent, it occurred to me to check with Wolfram Community -- and I'm glad I did! I had twisted around the #1 and #2 arguments.
POSTED BY: Bruce Colletti
POSTED BY: Jose Martin-Garcia
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