This post is a generalization of Union of the conjugates of a proper subgroup.
Consider an interval $[H,G]$ in the subgroup lattice of the finite group $G$, with $H \neq G$ and such that:
- (1) $ \exists x \in G$ with $\langle H,x \rangle = G$,
- (2) $\forall K \in [H,G]$ and $\forall g \in G$ we have $KgH = HgK$.
Remark: (1) is equivalent to: "the union of the coatoms of $[H,G]$ is a proper subset of $G$".
Definition: a coatom of $[H,G]$ is a maximal subgroup of $G$ containing $H$.
Question: Is the union of the conjugates of the coatoms of $[H,G]$, a proper subset of $G$?
Or equivalently: Is there $x \in G$ satisfying $\langle H,gxg^{-1} \rangle = G$ for all $g \in G$?
GAP checking: for $|G : H| < 32$ and $|G| \le 10^5$, there are (up to equivalence) $25608$ intervals $[H,G]$; $21918$ of them satisfy (1), $23323$ of them satisfy (2). Lastly, $21274$ of them satisfy both (1) and (2), and the question has a positive answer for all of them.
It is also true for any interval of finite groups $[H,G]$ with $G$ simple and $|G|<12180 = |A_1(29)|$.
This question generalizes the one cited above because that one reduces to have the proper subgroup to be maximal, but if $H$ is maximal then it satisfies the two above assumptions.
GAP code
IsNormalIntermediate:=function(G,H,K)
local D1,D2,s,i,j,E1,E2,c;
D1:=DoubleCosets(G,H,K);
s:=Size(DoubleCosets(G,H,K));
c:=0;
if s=Size(DoubleCosets(G,K,K)) then
return true;
else
D2:=DoubleCosets(G,K,H);
for i in [1..s] do
for j in [1..s] do
E1:=Elements(D1[i]);
E2:=Elements(D2[j]);
if Size(E1)=Size(E2) then
if E1=E2 then
c:=c+1;
fi;
fi;
od;
od;
fi;
return c=s;
end;;
IsDedekindInclusion:= function(G,H)
local K;
for K in IntermediateSubgroups(G,H).subgroups do
if not IsNormalIntermediate(G,H,K) then
return false;
fi;
od;
return true;
end;;
IsHcyclicInclusion:=function(G,H)
local D,dd,a;
D:=DoubleCosets(G,H,H);
a:=0;
for dd in D do
if Group(Set(dd))=G then
a:=1;
break;
fi;
od;
return a=1;
end;;
IsStronglyHcyclicInclusion:=function(G,H)
local int,sub,inc,c,i,j,M,m,U;
int:=IntermediateSubgroups(G,H);
sub:=int.subgroups; inc:=int.inclusions;
c:=Length(sub)+1;
if c=1 then
return true; # the maximal case is already known to be true
else
M:=List(Filtered(inc,i->i[2]=c),j->j[1]);
U:=Set(Union(List(M,m->Union(ConjugateSubgroups(G,sub[m])))));
fi;
if U<>Set(G) then
return true;
else
return false;
fi;
end;;
HcyclicNotStronglyDedekind:=function(d1,d2,o1,o2)
local n,d,r,o,G,H;
for d in [d1..d2] do
n:=NrTransitiveGroups(d);
for r in [1..n] do
G:=TransitiveGroup(d,r);
H:=Stabilizer(G,1);
o:=Order(G);
if o1<=o and o<=o2 then
if IsHcyclicInclusion(G,H) and IsDedekindInclusion(G,H) then
if not IsStronglyHcyclicInclusion(G,H) then
Print([d,r]);
fi;
fi;
fi;
od;
od;
end;;