GAP/Magma-cas: Suppose $H<S_n$ (given by generators): Does either system make it easy to find the maximal...
$begingroup$
I am not sure that this is the right forum, but anyhow:
Suppose I have a subgroup $H$ of $S_n$ (given by generators). Does either system make it easy to find the maximal subgroup containing $H$?
group-theory symmetric-groups gap magma-cas maximal-subgroup
edited Nov 28 '18 at 23:06
Shaun
8,832113681
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
$endgroup$
|
show 1 more comment
$begingroup$
I am not sure that this is the right forum, but anyhow:
Suppose I have a subgroup $H$ of $S_n$ (given by generators). Does either system make it easy to find the maximal subgroup containing $H$?
group-theory symmetric-groups gap magma-cas maximal-subgroup
edited Nov 28 '18 at 23:06
Shaun
8,832113681
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
$endgroup$
$begingroup$
"the"? ${}{}{}$
$endgroup$
– Dustan Levenstein
Jan 1 '17 at 14:44
$begingroup$
what does maximal mean here? a maximal subgroup which happens to contain $H$ or a subgroup that is maximal among those containing $H$?
$endgroup$
– Jorge Fernández
Jan 1 '17 at 14:47
$begingroup$
@JorgeFernándezHidalgo A $K$ such that $K$ is maximal in $S_n$ and contains $H.$
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@DustanLevenstein ?
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
2
$begingroup$
"the maximal subgroup" implies uniqueness, but there will normally be more than one. So do you want just one of them or all of them?
$endgroup$
– Derek Holt
Jan 1 '17 at 15:17
|
show 1 more comment
$begingroup$
I am not sure that this is the right forum, but anyhow:
Suppose I have a subgroup $H$ of $S_n$ (given by generators). Does either system make it easy to find the maximal subgroup containing $H$?
group-theory symmetric-groups gap magma-cas maximal-subgroup
edited Nov 28 '18 at 23:06
Shaun
8,832113681
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
$endgroup$
I am not sure that this is the right forum, but anyhow:
Suppose I have a subgroup $H$ of $S_n$ (given by generators). Does either system make it easy to find the maximal subgroup containing $H$?
group-theory symmetric-groups gap magma-cas maximal-subgroup
group-theory symmetric-groups gap magma-cas maximal-subgroup
edited Nov 28 '18 at 23:06
Shaun
8,832113681
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
edited Nov 28 '18 at 23:06
Shaun
8,832113681
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
edited Nov 28 '18 at 23:06
Shaun
8,832113681
edited Nov 28 '18 at 23:06
Shaun
8,832113681
edited Nov 28 '18 at 23:06
Shaun
8,832113681
8,832113681
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
asked Jan 1 '17 at 14:37
Igor RivinIgor Rivin
16k11234
16k11234
$begingroup$
"the"? ${}{}{}$
$endgroup$
– Dustan Levenstein
Jan 1 '17 at 14:44
$begingroup$
what does maximal mean here? a maximal subgroup which happens to contain $H$ or a subgroup that is maximal among those containing $H$?
$endgroup$
– Jorge Fernández
Jan 1 '17 at 14:47
$begingroup$
@JorgeFernándezHidalgo A $K$ such that $K$ is maximal in $S_n$ and contains $H.$
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@DustanLevenstein ?
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
2
$begingroup$
"the maximal subgroup" implies uniqueness, but there will normally be more than one. So do you want just one of them or all of them?
$endgroup$
– Derek Holt
Jan 1 '17 at 15:17
|
show 1 more comment
$begingroup$
"the"? ${}{}{}$
$endgroup$
– Dustan Levenstein
Jan 1 '17 at 14:44
$begingroup$
what does maximal mean here? a maximal subgroup which happens to contain $H$ or a subgroup that is maximal among those containing $H$?
$endgroup$
– Jorge Fernández
Jan 1 '17 at 14:47
$begingroup$
@JorgeFernándezHidalgo A $K$ such that $K$ is maximal in $S_n$ and contains $H.$
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@DustanLevenstein ?
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
2
$begingroup$
"the maximal subgroup" implies uniqueness, but there will normally be more than one. So do you want just one of them or all of them?
$endgroup$
– Derek Holt
Jan 1 '17 at 15:17
$begingroup$
"the"? ${}{}{}$
$endgroup$
– Dustan Levenstein
Jan 1 '17 at 14:44
$begingroup$
"the"? ${}{}{}$
$endgroup$
– Dustan Levenstein
Jan 1 '17 at 14:44
$begingroup$
what does maximal mean here? a maximal subgroup which happens to contain $H$ or a subgroup that is maximal among those containing $H$?
$endgroup$
– Jorge Fernández
Jan 1 '17 at 14:47
$begingroup$
what does maximal mean here? a maximal subgroup which happens to contain $H$ or a subgroup that is maximal among those containing $H$?
$endgroup$
– Jorge Fernández
Jan 1 '17 at 14:47
$begingroup$
@JorgeFernándezHidalgo A $K$ such that $K$ is maximal in $S_n$ and contains $H.$
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@JorgeFernándezHidalgo A $K$ such that $K$ is maximal in $S_n$ and contains $H.$
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@DustanLevenstein ?
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@DustanLevenstein ?
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
2
2
$begingroup$
"the maximal subgroup" implies uniqueness, but there will normally be more than one. So do you want just one of them or all of them?
$endgroup$
– Derek Holt
Jan 1 '17 at 15:17
$begingroup$
"the maximal subgroup" implies uniqueness, but there will normally be more than one. So do you want just one of them or all of them?
$endgroup$
– Derek Holt
Jan 1 '17 at 15:17
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
(I presume you want the maximal subgroup$color{red}{s}$ of $S_n$ containing $H$.)
You probably would have to compute at least part of the maximal subgroups (which is easy for $S_n$ if the degree is not too big) and then test conjugates of which subgroup (representative) contain $H$.
For example -- modifying the code for IntermediateSubgroups that is to be in the next release of GAP -- the following routine does this:
# ContainingMaximals(<group>,<sub>) returns all maximal subgroups of <group>
# that contain sub
ContainingMaximals:=function(G,U)
local uind,subs,incl,i,j,k,m,gens,t,c,p;
subs:=;
gens:=SmallGeneratingSet(U);
# find all maximals containing U
m:=MaximalSubgroupClassReps(G);
m:=Filtered(m,x->IndexNC(G,U) mod IndexNC(G,x)=0);
for j in m do
t:=RightTransversal(G,Normalizer(G,j)); # conjugates
for k in t do
if ForAll(gens,x->k*x/k in j) then
# U is contained in j^k
c:=j^k;
Assert(1,IsSubset(c,U));
Add(subs,c);
fi;
od;
od;
# rearrange
c:=List(subs,x->IndexNC(x,U));
p:=Sortex(c);
subs:=Permuted(subs,p);
return subs;
end;
This is not particularly clever standard code which nevertheless might be sufficient for doing a concrete example.
(I believe Magma has a variant of LowIndexSubgroups for permutation groups and that function might allow you to specify a subgroup that is to be contained, but I do not know that system enough to give details.)
edited Nov 28 '18 at 23:07
Shaun
8,832113681
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
votes
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oldest
votes
$begingroup$
(I presume you want the maximal subgroup$color{red}{s}$ of $S_n$ containing $H$.)
You probably would have to compute at least part of the maximal subgroups (which is easy for $S_n$ if the degree is not too big) and then test conjugates of which subgroup (representative) contain $H$.
For example -- modifying the code for IntermediateSubgroups that is to be in the next release of GAP -- the following routine does this:
# ContainingMaximals(<group>,<sub>) returns all maximal subgroups of <group>
# that contain sub
ContainingMaximals:=function(G,U)
local uind,subs,incl,i,j,k,m,gens,t,c,p;
subs:=;
gens:=SmallGeneratingSet(U);
# find all maximals containing U
m:=MaximalSubgroupClassReps(G);
m:=Filtered(m,x->IndexNC(G,U) mod IndexNC(G,x)=0);
for j in m do
t:=RightTransversal(G,Normalizer(G,j)); # conjugates
for k in t do
if ForAll(gens,x->k*x/k in j) then
# U is contained in j^k
c:=j^k;
Assert(1,IsSubset(c,U));
Add(subs,c);
fi;
od;
od;
# rearrange
c:=List(subs,x->IndexNC(x,U));
p:=Sortex(c);
subs:=Permuted(subs,p);
return subs;
end;
This is not particularly clever standard code which nevertheless might be sufficient for doing a concrete example.
(I believe Magma has a variant of LowIndexSubgroups for permutation groups and that function might allow you to specify a subgroup that is to be contained, but I do not know that system enough to give details.)
edited Nov 28 '18 at 23:07
Shaun
8,832113681
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
$endgroup$
add a comment |
$begingroup$
(I presume you want the maximal subgroup$color{red}{s}$ of $S_n$ containing $H$.)
You probably would have to compute at least part of the maximal subgroups (which is easy for $S_n$ if the degree is not too big) and then test conjugates of which subgroup (representative) contain $H$.
For example -- modifying the code for IntermediateSubgroups that is to be in the next release of GAP -- the following routine does this:
# ContainingMaximals(<group>,<sub>) returns all maximal subgroups of <group>
# that contain sub
ContainingMaximals:=function(G,U)
local uind,subs,incl,i,j,k,m,gens,t,c,p;
subs:=;
gens:=SmallGeneratingSet(U);
# find all maximals containing U
m:=MaximalSubgroupClassReps(G);
m:=Filtered(m,x->IndexNC(G,U) mod IndexNC(G,x)=0);
for j in m do
t:=RightTransversal(G,Normalizer(G,j)); # conjugates
for k in t do
if ForAll(gens,x->k*x/k in j) then
# U is contained in j^k
c:=j^k;
Assert(1,IsSubset(c,U));
Add(subs,c);
fi;
od;
od;
# rearrange
c:=List(subs,x->IndexNC(x,U));
p:=Sortex(c);
subs:=Permuted(subs,p);
return subs;
end;
This is not particularly clever standard code which nevertheless might be sufficient for doing a concrete example.
(I believe Magma has a variant of LowIndexSubgroups for permutation groups and that function might allow you to specify a subgroup that is to be contained, but I do not know that system enough to give details.)
edited Nov 28 '18 at 23:07
Shaun
8,832113681
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
$endgroup$
add a comment |
$begingroup$
(I presume you want the maximal subgroup$color{red}{s}$ of $S_n$ containing $H$.)
You probably would have to compute at least part of the maximal subgroups (which is easy for $S_n$ if the degree is not too big) and then test conjugates of which subgroup (representative) contain $H$.
For example -- modifying the code for IntermediateSubgroups that is to be in the next release of GAP -- the following routine does this:
# ContainingMaximals(<group>,<sub>) returns all maximal subgroups of <group>
# that contain sub
ContainingMaximals:=function(G,U)
local uind,subs,incl,i,j,k,m,gens,t,c,p;
subs:=;
gens:=SmallGeneratingSet(U);
# find all maximals containing U
m:=MaximalSubgroupClassReps(G);
m:=Filtered(m,x->IndexNC(G,U) mod IndexNC(G,x)=0);
for j in m do
t:=RightTransversal(G,Normalizer(G,j)); # conjugates
for k in t do
if ForAll(gens,x->k*x/k in j) then
# U is contained in j^k
c:=j^k;
Assert(1,IsSubset(c,U));
Add(subs,c);
fi;
od;
od;
# rearrange
c:=List(subs,x->IndexNC(x,U));
p:=Sortex(c);
subs:=Permuted(subs,p);
return subs;
end;
This is not particularly clever standard code which nevertheless might be sufficient for doing a concrete example.
(I believe Magma has a variant of LowIndexSubgroups for permutation groups and that function might allow you to specify a subgroup that is to be contained, but I do not know that system enough to give details.)
edited Nov 28 '18 at 23:07
Shaun
8,832113681
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
$endgroup$
(I presume you want the maximal subgroup$color{red}{s}$ of $S_n$ containing $H$.)
You probably would have to compute at least part of the maximal subgroups (which is easy for $S_n$ if the degree is not too big) and then test conjugates of which subgroup (representative) contain $H$.
For example -- modifying the code for IntermediateSubgroups that is to be in the next release of GAP -- the following routine does this:
# ContainingMaximals(<group>,<sub>) returns all maximal subgroups of <group>
# that contain sub
ContainingMaximals:=function(G,U)
local uind,subs,incl,i,j,k,m,gens,t,c,p;
subs:=;
gens:=SmallGeneratingSet(U);
# find all maximals containing U
m:=MaximalSubgroupClassReps(G);
m:=Filtered(m,x->IndexNC(G,U) mod IndexNC(G,x)=0);
for j in m do
t:=RightTransversal(G,Normalizer(G,j)); # conjugates
for k in t do
if ForAll(gens,x->k*x/k in j) then
# U is contained in j^k
c:=j^k;
Assert(1,IsSubset(c,U));
Add(subs,c);
fi;
od;
od;
# rearrange
c:=List(subs,x->IndexNC(x,U));
p:=Sortex(c);
subs:=Permuted(subs,p);
return subs;
end;
This is not particularly clever standard code which nevertheless might be sufficient for doing a concrete example.
(I believe Magma has a variant of LowIndexSubgroups for permutation groups and that function might allow you to specify a subgroup that is to be contained, but I do not know that system enough to give details.)
edited Nov 28 '18 at 23:07
Shaun
8,832113681
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
edited Nov 28 '18 at 23:07
Shaun
8,832113681
edited Nov 28 '18 at 23:07
Shaun
8,832113681
edited Nov 28 '18 at 23:07
Shaun
8,832113681
8,832113681
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
answered Jan 1 '17 at 17:26
ahulpkeahulpke
7,070926
7,070926
add a comment |
add a comment |
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$begingroup$
"the"? ${}{}{}$
$endgroup$
– Dustan Levenstein
Jan 1 '17 at 14:44
$begingroup$
what does maximal mean here? a maximal subgroup which happens to contain $H$ or a subgroup that is maximal among those containing $H$?
$endgroup$
– Jorge Fernández
Jan 1 '17 at 14:47
$begingroup$
@JorgeFernándezHidalgo A $K$ such that $K$ is maximal in $S_n$ and contains $H.$
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
$begingroup$
@DustanLevenstein ?
$endgroup$
– Igor Rivin
Jan 1 '17 at 14:58
2
$begingroup$
"the maximal subgroup" implies uniqueness, but there will normally be more than one. So do you want just one of them or all of them?
$endgroup$
– Derek Holt
Jan 1 '17 at 15:17