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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