I looked at
GAP maybe the title Computation of Hall Subgroup (given in previous year proposal) disguiding I mean that HallSubgroup( G, P ): computes a P-Hall subgroup for a set P of primes. This is a subgroup the order of which is only divisible by primes in P and whose index is coprime to all primes in P.
will HallSubgroup( G, P ) usefull for sympy?
I don't know..
Should I leave it for now?
Please see it's not a fully detailed we can break this down but it need discussion
Hall_Subgroup(G,P):
1.) Check if every p in P is prime if not:
valueerror
2.) Check if order of G is divisible by every prime in P if not:
return trivial subgroup (only Identity Element)
3.) Compute sylow p subgroup for every prime in P.
4.) Find the common of sylow p subgroups computed in above step.(since every sylow subgroup of a group is hallsubgroup)
This is not necessary. The order of a Hall subgroup does not have to be divisible be p if the order of the group is not.
Find the common of sylow p subgroups
A Hall subgroup must contain a Sylow p-group for each p in P.
suppose
P = [2,3] should'nt be hall subgroup be intersection of sylow 2 subgroup and sylow 3 subgroup for any group G.just clarifying .
I just noticed Probably If H is already a
Hall subgroup for a set of primes P then we can say that it is join of sylow p subgroup for p in P but it is not true vice versa. I have to think more for devising a algo.
suppose
P = [2,3]should'nt be hall subgroup be intersection ofsylow 2 subgroupandsylow 3 subgroupfor any group G.just clarifying .
I was completely wrong
So should I leave it ?
I am elobarating few of the ideas that I mentioned previously:
- Extend functionalities of polycyclic groups
There are various things that can be implemented such as:
1.)canonical polycyclic sequence for subgroup.
The pseudocode or Implementation of this algorithm is given in handbook and this is usefull in checking whether two subgroups of polycyclic presented group G are equal since every subgroup has unique canonical polycyclic sequence.This will be use as a helper method for the below method.
2.)check if two subgroups of a polycyclic presented groups are equal or not.
3.)polycyclic orbit-stabilizer algorithm:
This algorithm is also given in the handbook with detailed implementation.
4.)numerator pcgs
5.)denominator pcgs
both numerator and denominator pcgs I have taken reference from the gap(https://www.biu.ac.il/os_site/documentation/gap4r1/ref/CHAP040.htm#SECT011) 40.9
- Automorphism groups for finite solvable groups
I have started reading automorphism part from handbook chapter - 8 and also find the thesis with implentation of some usefull algorithms here (https://openresearch-repository.anu.edu.au/bitstream/1885/133102/4/b19051244_Smith_Michael_J.pdf)
- Quotient groups
As you have asked for what kind of groups I think for Finitely presented group (Fp group) In the chapter 9 of the handbook various algorithms are described which can be usefull.
some algorithms such as epimorphism can also be used for computing automorphism group.
I am still working for more descriptive description (As I am still reading the book) And btw I am thinking to start working on canonical polycyclic sequence.
Addition of method to compute
for a reference
homomorphism from one polycyclic group G to another polycyclic group H would be usefull?for a reference
gap has a method for it where gens is the list of generators of G and imgs are the list of images of elements of H.GroupHomomorphismByImages(G, H, gens, imgs)
just asking will that work for
polycyclic group? beacuse the method we have now is checking whether the group G and H are PermutationGroup, FpGroup, FreeGroup if not isinstance(domain, (PermutationGroup, FpGroup, FreeGroup)):
raise TypeError("The domain must be a group")
if not isinstance(codomain, (PermutationGroup, FpGroup, FreeGroup)):
raise TypeError("The codomain must be a group")
After doing some research I have find some implementation of pseudo code for them.
I looked at the algorithm describes to see the
Here is the algorithm (http://huonw.github.io/honours-thesis.pdf)
difficulty of algorithm I find many of the required sub function already implemented in sympy.I am majorly concerned about the step 10 of the algorithm.Here is the algorithm (http://huonw.github.io/honours-thesis.pdf)
page no. 22.
However theres also a naive algorithm which is very easy to implement it is also described in the book as follows:
socle is generated by minimal normal subgroups, one approach to finding it would be enumerating the normal subgroups, recording those that are minimal and then taking the subgroup that they generate. This could be done by iterating over group elements z ∈ G and taking the normal closures ⟨z^G⟩: every
minimal normal subgroup is sure to appear in this list