I think there is an existing method in SymPy to check the normality of permutation groups.
Yes It is there as is_normal.
Hello Kalevi,
Good morning I was thinking about adding a functions to check whether a group is a hall subgroups or not (https://en.wikipedia.org/wiki/Hall_subgroup.
Since we have all the required method that we need to implement a test for hall subgroups it will not take much time.
this is the possible algorithm I am thinking:
is_hall_subgroups(H,G):
#We want to check if H is a hall subgroups and H is the subgroup of G where G should be a finite group.
1.)check if G is a finite groups
2.)find the order of H.(Ord)
3.)find the index for H.(ind)
4.)If gcd(Ord,Ind) is 1 then H is a hall_subgroup.(gcd = Greatest Common Divisor)
Sorry for late reply
I suggest this because there was a Computation of Hall Subgroup idea proposed in last year gsoc proposal but due to time constraint it was not implemented So I was looking to implement that but while looking I found this hall subgroup property so i was thinking it may be helpfull to add it but after a thorough research I didnt find it helpfull.
do we also need Computation of Hall Subgroup ?
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?
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)Find the common of sylow p subgroups
P = [2,3] should'nt be hall subgroup be intersection of sylow 2 subgroup and sylow 3 subgroup for any group G.just clarifying .
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
- 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
- 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.
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)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")