[GAP Forum] About permutable

[Benjamin Sambale]

Dear Jianjun,

you may try this code (no warranty):

prod:=function(x,y) return x*y; end;

IsPermutable:=function(G,H)
local g;
     if IsNormal(G,H) then return true; fi;
     if ForAll(Set(G),g->SetX(H,Group(g),prod)=SetX(Group(g),H,prod)) 
then return true; else return false; fi;
end;

AllSgPermutable:=function(G)
     if not IsNilpotentGroup(G) then return false; fi;
     if 
ForAll(List(ConjugacyClassesSubgroups(G),Representative),H->IsPermutable(G,H)) 
then return true; else return false; fi;
end;

Here observe that it is only necessary to test permutability with cyclic 
subgroups. Moreover, if all subgroups are permutable in a finite group 
G, then G must be nilpotent, since two different Sylow p-subgroups (for 
the same prime p) are certainly not permutable.

Best wishes,
Benjamin Sambale

Am 13.07.2011 16:28, schrieb 刘建军:
> Dear forum,
>
> A subgroup H of a finite group G is said to be permutable if HK=KH for every subgroup K of G.
>
> I would like to know whether all subgroups of a group G are permutable.
> Is there a method to get it in GAP?
>
> Best Wishes
> Jianjun Liu
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