# Maths - Program

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 ```

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.