4
$\begingroup$

I am new to Mathematica, and I would love to know if there is a function that that returns the double cosets of a group $G$ w.r.t a subgroup $H$, where a double coset is defined as $HxH=\{h_1xh_2,\forall h_1,h_2\in H\}$, for some $x$ in $G$. In particular, I am mostly concerned with the the number of such cosets, so it is also satisifiying if there is a function for such countings. If there's no such general functions, is there special cases for $G$ a symmetric group?

$\endgroup$
1
  • 3
    $\begingroup$ Arguments about the votes to close this question and opinions about leaving it open are discussed here. $\endgroup$
    – rhermans
    May 17, 2022 at 9:32

1 Answer 1

7
$\begingroup$

Of course MMA don't have any of the functions you need, but we can get the double cosets of a group $G$ w.r.t a subgroup $H$ by it. Such as $S_5$. As we know, $D_5$ is a subgroup of it. Then we can get its double cosets like:

G = SymmetricGroup[5];
H = DihedralGroup[5];

Sort[DeleteDuplicates[Flatten[Outer[PermutationProduct, GroupElements[H], 
    GroupElements[G], GroupElements[H]]]], Less]
TemplateApply["The order of H is <*Length[%]*>"]

enter image description here

In particular, I am mostly concerned with the number of such cosets

I'm not sure I understand this statement of yours. If by definition, any subgroup of G can generate double cosets, so you're actually asking how many subgroups of G are there? For the $S_5$:

FiniteGroupData[{"SymmetricGroup", 5}, "Subgroups"]

{Trivial,{CyclicGroup,2},{CyclicGroup,3},{CyclicGroup,4},Vierergruppe,{CyclicGroup,5},{CyclicGroup,6},{SymmetricGroup,3},{DihedralGroup,4},{DihedralGroup,5},{AlternatingGroup,4},{DihedralGroup,6},{SemidirectProduct,{{CyclicGroup,4},{CyclicGroup,5}}},{SymmetricGroup,4},{AlternatingGroup,5},{SymmetricGroup,5}}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.