There are a number of open source programs that can work with groups. I have used Axiom, how to install Axiom here.
I have put user input in red:
(1) -> genA6 : LIST PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]] (1) [(1 2 3),(2 3 4 5 6)] Type: List Permutation Integer (2) -> pRA6 := permutationRepresentation(genA6,6) +0 0 1 0 0 0+ +1 0 0 0 0 0+ | | | | |1 0 0 0 0 0| |0 0 0 0 0 1| | | | | |0 1 0 0 0 0| |0 1 0 0 0 0| (2) [| |,| |] |0 0 0 1 0 0| |0 0 1 0 0 0| | | | | |0 0 0 0 1 0| |0 0 0 1 0 0| | | | | +0 0 0 0 0 1+ +0 0 0 0 1 0+ Type: List Matrix Integer (3) -> sp0 := meatAxe(pRA6::(LIST MATRIX PF 2)) A proper cyclic submodule is found. Transition matrix computed The inverse of the transition matrix computed Now transform the matrices +0 0 1 0 0+ +1 0 0 0 0+
| | | |
|1 0 0 0 0| |1 1 1 1 1|
| | | |
(3) [[|0 1 0 0 0|,|0 1 0 0 0|],[[1],[1]]]
| | | |
|0 0 0 1 0| |0 0 1 0 0|
| | | |
+0 0 0 0 1+ +0 0 0 1 0+
Type: List List Matrix PrimeField 2
(4) -> sp1 := meatAxe sp0.1
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
We know that all the cyclic submodules generated by all
non-trivial element of the singular matrix under view are
not proper, hence Norton's irreducibility test can be done:
A proper cyclic submodule is found.
Transition matrix computed
The inverse of the transition matrix computed
Now transform the matrices
Representation is not irreducible and it will be split:
+0 1 0 0+ +0 1 1 1+
| | | |
|0 0 1 0| |1 1 0 1|
(4) [[[1],[1]],[| |,| |]]
|1 0 0 0| |1 1 1 0|
| | | |
+0 0 0 1+ +1 1 1 1+
Type: List List Matrix PrimeField 2
(5) -> isAbsolutelyIrreducible? sp1.2
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra has
one-dimensional kernel
We know that all the cyclic submodules generated by all
non-trivial element of the singular matrix under view are
not proper, hence Norton's irreducibility test can be done:
The generated cyclic submodule was not proper
Representation is absolutely irreducible
(5) true
Type: Boolean
(6) -> d2211 := irreducibleRepresentation([2,2,1,1],genA6)
(6)
+1 0 0 - 1 1 0 0 0 0 + + 0 0 1 0 0 0 1 0 0+
| | | |
|0 1 0 1 0 1 0 0 0 | | 0 0 0 0 1 0 - 1 0 0|
| | | |
|0 0 1 0 1 - 1 0 0 0 | | 0 0 0 0 0 1 1 0 0|
| | | |
|0 0 0 - 1 0 0 - 1 0 0 | | 0 0 0 0 0 0 1 1 0|
| | | |
[|0 0 0 0 - 1 0 0 - 1 0 |,| 0 0 0 0 0 0 - 1 0 1|]
| | | |
|0 0 0 0 0 - 1 0 0 - 1| | 0 0 0 0 0 0 1 0 0|
| | | |
|0 0 0 1 0 0 0 0 0 | |- 1 0 0 0 0 0 - 1 0 0|
| | | |
|0 0 0 0 1 0 0 0 0 | | 0 - 1 0 0 0 0 1 0 0|
| | | |
+0 0 0 0 0 1 0 0 0 + + 0 0 0 - 1 0 0 - 1 0 0+
Type: List Matrix Integer
(7) -> d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2
Fingerprint element in generated algebra is singular
A proper cyclic submodule is found.
Transition matrix computed
The inverse of the transition matrix computed
Now transform the matrices
+1 0 0 0 0+ +1 1 1 0 0+
+1 0 1 1+ +0 0 1 0+ | | | |
| | | | |0 1 1 1 1| |0 0 1 1 1|
|0 1 0 1| |1 1 1 1| | | | |
(7) [[| |,| |],[|0 1 1 0 0|,|1 0 0 1 0|]]
|1 1 0 0| |1 0 1 1| | | | |
| | | | |0 1 0 1 0| |0 0 1 0 1|
+0 1 0 0+ +0 1 0 1+ | | | |
+0 1 1 1 0+ +1 0 0 1 1+
Type: List List Matrix PrimeField 2
(8) -> isAbsolutelyIrreducible? sp2.1
Random element in generated algebra has
one-dimensional kernel
We know that all the cyclic submodules generated by all
non-trivial element of the singular matrix under view are
not proper, hence Norton's irreducibility test can be done:
The generated cyclic submodule was not proper
Representation is absolutely irreducible
(8) true
Type: Boolean
(9) -> areEquivalent? (sp1.2,sp2.1)
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Dimensions of kernels differ
Representations are not equivalent.
(9) [0]
Type: Matrix PrimeField 2
(10) -> dA6d16 := tensorProduct(sp1.2,sp2.1);meatAxe dA6d16
Fingerprint element in generated algebra is non-singular
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
Fingerprint element in generated algebra is non-singular
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
We know that all the cyclic submodules generated by all
non-trivial element of the singular matrix under view are
not proper, hence Norton's irreducibility test can be done:
The generated cyclic submodule was not proper
Representation is irreducible, but we don't know
whether it is absolutely irreducible
(10)
[
+0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0+
| |
|0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0|
| |
|1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
[| |,
|0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0|
| |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1|
| |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0|
| |
+0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0+
+0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0+
| |
|0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1|
| |
|0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0|
| |
|0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1|
| |
|0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0|
| |
|0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1|
| |
|1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0|
| |
|0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1|
| |]
|0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0|
| |
|0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1|
| |
|1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0|
| |
|0 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1|
| |
|0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0|
| |
|0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1|
| |
|1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0|
| |
+0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1+
]
Type: List List Matrix PrimeField 2
(11) -> isAbsolutelyIrreducible? dA6d16
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
We have not found a one-dimensional kernel so far,
as we do a random search you could try again
(11) false
Type: Boolean
(12) -> sp3 := meatAxe (dA6d16:: (LIST MATRIX FF(2,2)))
Fingerprint element in generated algebra is non-singular
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
Fingerprint element in generated algebra is non-singular
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
Fingerprint element in generated algebra is singular
The generated cyclic submodule was not proper
The generated cyclic submodule was not proper
A proper cyclic submodule is found.
Transition matrix computed
The inverse of the transition matrix computed
Now transform the matrices
(12) [ + %A %A + 1 0 %A 1 %A + 1 0 0 + | | | 0 0 %A %A + 1 %A %A 0 0 | | | | %A %A + 1 %A 1 %A + 1 0 0 0 | | | | %A %A + 1 %A 1 %A 0 0 0 | [| |, |%A + 1 1 1 1 0 0 %A + 1 %A| | | | 0 0 %A + 1 1 0 0 %A 0 | | | | 1 0 1 1 0 0 0 0 | | | + 1 1 0 0 0 0 0 0 + + 1 0 %A 0 1 1 %A %A + 1+ | | | 1 %A + 1 0 0 0 %A + 1 1 %A + 1| | | | %A 1 %A + 1 %A + 1 %A + 1 1 %A 0 | | | |%A + 1 %A + 1 0 0 1 %A + 1 1 1 | | |] | 1 0 %A + 1 0 1 1 %A %A | | | | 0 0 %A + 1 %A + 1 %A + 1 1 1 %A | | | | 0 0 1 0 0 1 0 1 | | | + 0 %A 0 %A 1 %A + 1 %A + 1 %A + , +0 1 1 %A + 1 0 0 0 0+
| |
|1 1 %A + 1 0 0 0 0 0|
| |
|%A 0 0 0 0 0 0 0|
| |
|1 %A 0 0 0 0 0 0|
[| |,
|%A %A + 1 1 1 1 0 1 1|
| |
|0 0 %A 1 0 1 0 1|
| |
|%A 1 0 1 1 1 0 0|
| |
+1 %A %A + 1 %A 0 1 0 0+
+%A + 1 1 %A 0 0 %A + 1 0 1 +
| |
| 0 %A 1 1 1 0 %A + 1 %A |
| |
| 0 %A + 1 0 %A + 1 %A + 1 1 %A + 1 %A |
| |
| 1 %A + 1 1 %A + 1 0 0 %A + 1 1 |
| |]
| 0 %A 0 %A + 1 %A + 1 0 0 %A + 1|
| |
|%A + 1 0 %A + 1 %A 0 %A + 1 0 %A + 1|
| |
| 0 1 0 1 %A + 1 0 %A + 1 %A + 1|
| |
+ %A %A %A 1 %A %A 1 %A + 1+
]
Type: List List Matrix FiniteField(2,2)
(13) -> isAbsolutelyIrreducible? sp3.1
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra has
one-dimensional kernel
We know that all the cyclic submodules generated by all
non-trivial element of the singular matrix under view are
not proper, hence Norton's irreducibility test can be done:
The generated cyclic submodule was not proper
Representation is absolutely irreducible
(13) true
Type: Boolean
(14) -> isAbsolutelyIrreducible? sp3.2
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra has
one-dimensional kernel
We know that all the cyclic submodules generated by all
non-trivial element of the singular matrix under view are
not proper, hence Norton's irreducibility test can be done:
The generated cyclic submodule was not proper
Representation is absolutely irreducible
(14) true
Type: Boolean
(15) -> areEquivalent? (sp3.1,sp3.2)
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra does
not have a one-dimensional kernel
Random element in generated algebra has
one-dimensional kernel
There is no isomorphism, as the only possible one
fails to do the necessary base change
Representations are not equivalent. (15) [0]
Type: Matrix FiniteField(2,2)
(16) -> sp0.2
(16) [[1],[1]]
Type: List Matrix PrimeField 2
(17) -> sp1.2
+0 1 0 0+ +0 1 1 1+
| | | |
|0 0 1 0| |1 1 0 1|
(17) [| |,| |]
|1 0 0 0| |1 1 1 0|
| | | |
+0 0 0 1+ +1 1 1 1+
Type: List Matrix PrimeField 2
(18) -> sp2.1
+1 0 1 1+ +0 0 1 0+
| | | |
|0 1 0 1| |1 1 1 1|
(18) [| |,| |]
|1 1 0 0| |1 0 1 1|
| | | |
+0 1 0 0+ +0 1 0 1+
Type: List Matrix PrimeField 2
(19) -> sp3.1
(19)
+ %A %A + 1 0 %A 1 %A + 1 0 0 +
| |
| 0 0 %A %A + 1 %A %A 0 0 |
| |
| %A %A + 1 %A 1 %A + 1 0 0 0 |
| |
| %A %A + 1 %A 1 %A 0 0 0 |
[| |,
|%A + 1 1 1 1 0 0 %A + 1 %A|
| |
| 0 0 %A + 1 1 0 0 %A 0 |
| |
| 1 0 1 1 0 0 0 0 |
| |
+ 1 1 0 0 0 0 0 0 +
+ 1 0 %A 0 1 1 %A %A + 1+
| |
| 1 %A + 1 0 0 0 %A + 1 1 %A + 1|
| |
| %A 1 %A + 1 %A + 1 %A + 1 1 %A 0 |
| |
|%A + 1 %A + 1 0 0 1 %A + 1 1 1 |
| |]
| 1 0 %A + 1 0 1 1 %A %A |
| |
| 0 0 %A + 1 %A + 1 %A + 1 1 1 %A |
| |
| 0 0 1 0 0 1 0 1 |
| |
+ 0 %A 0 %A 1 %A + 1 %A + 1 %A +
Type: List Matrix FiniteField(2,2)
(20) -> sp3.2
(20)
+0 1 1 %A + 1 0 0 0 0+
| |
|1 1 %A + 1 0 0 0 0 0|
| |
|%A 0 0 0 0 0 0 0|
| |
|1 %A 0 0 0 0 0 0|
[| |,
|%A %A + 1 1 1 1 0 1 1|
| |
|0 0 %A 1 0 1 0 1|
| |
|%A 1 0 1 1 1 0 0|
| |
+1 %A %A + 1 %A 0 1 0 0+
+%A + 1 1 %A 0 0 %A + 1 0 1 +
| |
| 0 %A 1 1 1 0 %A + 1 %A |
| |
| 0 %A + 1 0 %A + 1 %A + 1 1 %A + 1 %A |
| |
| 1 %A + 1 1 %A + 1 0 0 %A + 1 1 |
| |]
| 0 %A 0 %A + 1 %A + 1 0 0 %A + 1|
| |
|%A + 1 0 %A + 1 %A 0 %A + 1 0 %A + 1|
| |
| 0 1 0 1 %A + 1 0 %A + 1 %A + 1|
| |
+ %A %A %A 1 %A %A 1 %A + 1+
Type: List Matrix FiniteField(2,2)
(21) -> dA6d16
(21)
+0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0+
| |
|0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0|
[| |,
|1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0|
| |
|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1|
| |
|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1|
| |
|0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0|
| |
+0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0+
+0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0+
| |
|0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1|
| |
|0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1|
| |
|0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1|
| |
|0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0|
| |
|1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1|
| |
|1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1|
| |
|0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1|
| |]
|0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0|
| |
|1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0|
| |
|1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0|
| |
|0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0|
| |
|0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0|
| |
|1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1|
| |
|1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1|
| |
+0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1+
Type: List Matrix PrimeField 2
|
