Mixing
How to do inflation of $k(G/N)$-modules with MAGMA?
I'd like to ask the following MAGMA-question:
Let $k$ be a field of characteristic $p$, $G$ be a finite group and $N$ a normal subgroup of $G$.
Moreover, let $M$ be a $k(G/N)$-module.
Is it possible with MAGMA to get the inflation, i.e. $M$ viewed as a $kG$-module?
I tried something below (for the special case where $M$ is a projective $k(G/N)$-module), but it didn't work, unfortunately.
G:=...
N:=...
K:=GF(16);
FAC,f:=quo<G|N>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
Nnew:=sub<G|[n:n in N]>;
M:=GModule(Nnew,[ u(g) : g in Nnew ]);
Any help is appreciated.
EDIT:
I looked at the following concrete example:
G:=Sym(5);
m:=Exponent(G);
Kx<x>:=PolynomialRing(GF(2));
f:=x^m-1;
L:=SplittingField(f);
u:=#L;
K:=GF(u);
P2:=sub<G|[(2,4)]>;
N2:=sub<G|[(2,4),(3,5),(1,3,5)]>;
P2inN2:=sub<N2|[(2,4)]>;
FAC,f:=quo<N2|P2inN2>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
N2new:=sub<G|[g:g in N2]>;
GGG:=Generators(N2new);
M:=GModule(N2new,[ u(g) : g in N2new ]);
I:=Induction(M,G);
dirI:=DirectSumDecomposition(I);
dirI;
But $I$ is decomposable over a splitting field (it decomposes into a direct sum of two indec. $kG$-modules having $k$-dimensions 4 and 16, respectively).
Note that the inflation of P2 is isomorphic to a simple kN2-module. When I induce this simple kN2-module to G, everything works.
group-theory finite-groups magma-cas
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
add a comment |
I'd like to ask the following MAGMA-question:
Let $k$ be a field of characteristic $p$, $G$ be a finite group and $N$ a normal subgroup of $G$.
Moreover, let $M$ be a $k(G/N)$-module.
Is it possible with MAGMA to get the inflation, i.e. $M$ viewed as a $kG$-module?
I tried something below (for the special case where $M$ is a projective $k(G/N)$-module), but it didn't work, unfortunately.
G:=...
N:=...
K:=GF(16);
FAC,f:=quo<G|N>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
Nnew:=sub<G|[n:n in N]>;
M:=GModule(Nnew,[ u(g) : g in Nnew ]);
Any help is appreciated.
EDIT:
I looked at the following concrete example:
G:=Sym(5);
m:=Exponent(G);
Kx<x>:=PolynomialRing(GF(2));
f:=x^m-1;
L:=SplittingField(f);
u:=#L;
K:=GF(u);
P2:=sub<G|[(2,4)]>;
N2:=sub<G|[(2,4),(3,5),(1,3,5)]>;
P2inN2:=sub<N2|[(2,4)]>;
FAC,f:=quo<N2|P2inN2>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
N2new:=sub<G|[g:g in N2]>;
GGG:=Generators(N2new);
M:=GModule(N2new,[ u(g) : g in N2new ]);
I:=Induction(M,G);
dirI:=DirectSumDecomposition(I);
dirI;
But $I$ is decomposable over a splitting field (it decomposes into a direct sum of two indec. $kG$-modules having $k$-dimensions 4 and 16, respectively).
Note that the inflation of P2 is isomorphic to a simple kN2-module. When I induce this simple kN2-module to G, everything works.
group-theory finite-groups magma-cas
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
1
An example that can actually be copy/pasted to run and troubleshoot would be helpful. When you say it "doesn't work", what sort of error message are you seeing?
– Morgan Rodgers
Nov 20 '18 at 20:17
Thank you very much for your comment. I edited the question.
– Bernhard Boehmler
Nov 20 '18 at 21:12
add a comment |
I'd like to ask the following MAGMA-question:
Let $k$ be a field of characteristic $p$, $G$ be a finite group and $N$ a normal subgroup of $G$.
Moreover, let $M$ be a $k(G/N)$-module.
Is it possible with MAGMA to get the inflation, i.e. $M$ viewed as a $kG$-module?
I tried something below (for the special case where $M$ is a projective $k(G/N)$-module), but it didn't work, unfortunately.
G:=...
N:=...
K:=GF(16);
FAC,f:=quo<G|N>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
Nnew:=sub<G|[n:n in N]>;
M:=GModule(Nnew,[ u(g) : g in Nnew ]);
Any help is appreciated.
EDIT:
I looked at the following concrete example:
G:=Sym(5);
m:=Exponent(G);
Kx<x>:=PolynomialRing(GF(2));
f:=x^m-1;
L:=SplittingField(f);
u:=#L;
K:=GF(u);
P2:=sub<G|[(2,4)]>;
N2:=sub<G|[(2,4),(3,5),(1,3,5)]>;
P2inN2:=sub<N2|[(2,4)]>;
FAC,f:=quo<N2|P2inN2>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
N2new:=sub<G|[g:g in N2]>;
GGG:=Generators(N2new);
M:=GModule(N2new,[ u(g) : g in N2new ]);
I:=Induction(M,G);
dirI:=DirectSumDecomposition(I);
dirI;
But $I$ is decomposable over a splitting field (it decomposes into a direct sum of two indec. $kG$-modules having $k$-dimensions 4 and 16, respectively).
Note that the inflation of P2 is isomorphic to a simple kN2-module. When I induce this simple kN2-module to G, everything works.
group-theory finite-groups magma-cas
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
I'd like to ask the following MAGMA-question:
Let $k$ be a field of characteristic $p$, $G$ be a finite group and $N$ a normal subgroup of $G$.
Moreover, let $M$ be a $k(G/N)$-module.
Is it possible with MAGMA to get the inflation, i.e. $M$ viewed as a $kG$-module?
I tried something below (for the special case where $M$ is a projective $k(G/N)$-module), but it didn't work, unfortunately.
G:=...
N:=...
K:=GF(16);
FAC,f:=quo<G|N>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
Nnew:=sub<G|[n:n in N]>;
M:=GModule(Nnew,[ u(g) : g in Nnew ]);
Any help is appreciated.
EDIT:
I looked at the following concrete example:
G:=Sym(5);
m:=Exponent(G);
Kx<x>:=PolynomialRing(GF(2));
f:=x^m-1;
L:=SplittingField(f);
u:=#L;
K:=GF(u);
P2:=sub<G|[(2,4)]>;
N2:=sub<G|[(2,4),(3,5),(1,3,5)]>;
P2inN2:=sub<N2|[(2,4)]>;
FAC,f:=quo<N2|P2inN2>;
PIMs:=ProjectiveIndecomposableModules(FAC,K);
P2:=PIMs[2];
phi2:=Representation(P2);
u:=f*phi2;
N2new:=sub<G|[g:g in N2]>;
GGG:=Generators(N2new);
M:=GModule(N2new,[ u(g) : g in N2new ]);
I:=Induction(M,G);
dirI:=DirectSumDecomposition(I);
dirI;
But $I$ is decomposable over a splitting field (it decomposes into a direct sum of two indec. $kG$-modules having $k$-dimensions 4 and 16, respectively).
Note that the inflation of P2 is isomorphic to a simple kN2-module. When I induce this simple kN2-module to G, everything works.
group-theory finite-groups magma-cas
group-theory finite-groups magma-cas
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
edited Nov 20 '18 at 21:30
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
asked Nov 20 '18 at 20:10
Bernhard Boehmler
422212
422212
1
An example that can actually be copy/pasted to run and troubleshoot would be helpful. When you say it "doesn't work", what sort of error message are you seeing?
– Morgan Rodgers
Nov 20 '18 at 20:17
Thank you very much for your comment. I edited the question.
– Bernhard Boehmler
Nov 20 '18 at 21:12
add a comment |
1
An example that can actually be copy/pasted to run and troubleshoot would be helpful. When you say it "doesn't work", what sort of error message are you seeing?
– Morgan Rodgers
Nov 20 '18 at 20:17
Thank you very much for your comment. I edited the question.
– Bernhard Boehmler
Nov 20 '18 at 21:12
1
1
An example that can actually be copy/pasted to run and troubleshoot would be helpful. When you say it "doesn't work", what sort of error message are you seeing?
– Morgan Rodgers
Nov 20 '18 at 20:17
An example that can actually be copy/pasted to run and troubleshoot would be helpful. When you say it "doesn't work", what sort of error message are you seeing?
– Morgan Rodgers
Nov 20 '18 at 20:17
Thank you very much for your comment. I edited the question.
– Bernhard Boehmler
Nov 20 '18 at 21:12
Thank you very much for your comment. I edited the question.
– Bernhard Boehmler
Nov 20 '18 at 21:12
add a comment |
1 Answer
1
active
oldest
votes
The problem here is that the list of elements
[g : g in N2]
is not coming out in the same order as the list
[g : g in N2new]
so the module M that you are constructing is not a valid module. (It would be too time consuming for Magma to attempt to check that the matrices that you specify really do define a module for the group.)
If instead you do
M:=GModule(N2new, [ u(g) : g in N2 ]);
then it will work.
But I don't really understand why you are taking every element of the group as a generator. That would be a very bad idea if the group was much larger. Why not define M as a module for N2, which you can do with:
M := GModule(N2, [ u(N2.i) : i in [1..Ngens(N2)] ] );
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
add a comment |
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1 Answer
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1 Answer
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oldest
votes
The problem here is that the list of elements
[g : g in N2]
is not coming out in the same order as the list
[g : g in N2new]
so the module M that you are constructing is not a valid module. (It would be too time consuming for Magma to attempt to check that the matrices that you specify really do define a module for the group.)
If instead you do
M:=GModule(N2new, [ u(g) : g in N2 ]);
then it will work.
But I don't really understand why you are taking every element of the group as a generator. That would be a very bad idea if the group was much larger. Why not define M as a module for N2, which you can do with:
M := GModule(N2, [ u(N2.i) : i in [1..Ngens(N2)] ] );
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
add a comment |
The problem here is that the list of elements
[g : g in N2]
is not coming out in the same order as the list
[g : g in N2new]
so the module M that you are constructing is not a valid module. (It would be too time consuming for Magma to attempt to check that the matrices that you specify really do define a module for the group.)
If instead you do
M:=GModule(N2new, [ u(g) : g in N2 ]);
then it will work.
But I don't really understand why you are taking every element of the group as a generator. That would be a very bad idea if the group was much larger. Why not define M as a module for N2, which you can do with:
M := GModule(N2, [ u(N2.i) : i in [1..Ngens(N2)] ] );
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
add a comment |
The problem here is that the list of elements
[g : g in N2]
is not coming out in the same order as the list
[g : g in N2new]
so the module M that you are constructing is not a valid module. (It would be too time consuming for Magma to attempt to check that the matrices that you specify really do define a module for the group.)
If instead you do
M:=GModule(N2new, [ u(g) : g in N2 ]);
then it will work.
But I don't really understand why you are taking every element of the group as a generator. That would be a very bad idea if the group was much larger. Why not define M as a module for N2, which you can do with:
M := GModule(N2, [ u(N2.i) : i in [1..Ngens(N2)] ] );
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
The problem here is that the list of elements
[g : g in N2]
is not coming out in the same order as the list
[g : g in N2new]
so the module M that you are constructing is not a valid module. (It would be too time consuming for Magma to attempt to check that the matrices that you specify really do define a module for the group.)
If instead you do
M:=GModule(N2new, [ u(g) : g in N2 ]);
then it will work.
But I don't really understand why you are taking every element of the group as a generator. That would be a very bad idea if the group was much larger. Why not define M as a module for N2, which you can do with:
M := GModule(N2, [ u(N2.i) : i in [1..Ngens(N2)] ] );
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
answered Nov 20 '18 at 21:44
Derek Holt
52.6k53570
52.6k53570
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
add a comment |
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
I'm relatively new to MAGMA and, unfortunately, I didn't know the command with u(N2.i) : i in[1..Ngens(N2)]. At first, I didn't take every element of the group as a generator, but the module didn't split. Then, I misunderstood something in the manual and wanted to try it this way, but, of course, it didn't change anything, but I printed the most recent version of my tries here. Thank you very much for your answer and the explanations! :-)
– Bernhard Boehmler
Nov 21 '18 at 11:15
add a comment |
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An example that can actually be copy/pasted to run and troubleshoot would be helpful. When you say it "doesn't work", what sort of error message are you seeing?
– Morgan Rodgers
Nov 20 '18 at 20:17
Thank you very much for your comment. I edited the question.
– Bernhard Boehmler
Nov 20 '18 at 21:12