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##### Activity
Mohit Gupta
@mohitacecode
But I wonder this may be usefull in other cases too? I don't have any examples now(I will look for it)
Divyanshu Thakur
@divyanshu132
I think there is an existing method in SymPy to check the normality of permutation groups.
Divyanshu Thakur
@divyanshu132
In addition to elaborating as Kalevi has pointed out, I would suggest you to get yourself familiar with the presentation of groups in SymPy may be you can try adding some small missing functionalities in that way you will have an idea of how much research it takes to add a small functionality . Eventually, you'll be able to figure it out how much of work can be done in a summer.
Mohit Gupta
@mohitacecode
okay I will first look for adding small functionalities maybe I will get familiar with the code much faster this way and will be able to estimate the right and sufficient amount of work for one summer.

I think there is an existing method in SymPy to check the normality of permutation groups.

Yes It is there as is_normal.

Mohit Gupta
@mohitacecode

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)
Kalevi Suominen
@jksuom
Have you some idea of where it would be useful to know that a subgroup is Hall subgroup?
Mohit Gupta
@mohitacecode

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 ?

Kalevi Suominen
@jksuom
Hall subgroups are subgroups that have a certain property (which is easy to check). There is no specific Hall subgroup that could be computed.
Mohit Gupta
@mohitacecode
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.
Mohit Gupta
@mohitacecode
Kalevi Suominen
@jksuom
That is what sylow_subgroup does if there is exactly one prime in P. So that looks like a generalization.
Mohit Gupta
@mohitacecode
just clarifying my doubt sylow_subgroup genrates a subgroup for any particular prime I think ?
Mohit Gupta
@mohitacecode

HallSubgroup( G, P ) will generate a subgroup for a list of primes. maybe we can use sylow_subgroup and take intersection(not there now in sympy) for them.

will HallSubgroup( G, P ) usefull for sympy?

Kalevi Suominen
@jksuom
One often speaks of Sylow p-subgroups for a particular prime.
will HallSubgroup( G, P ) usefull for sympy?
I don't know..
Mohit Gupta
@mohitacecode

One often speaks of Sylow p-subgroups for a particular prime.

Okay..

Should I leave it for now?
Kalevi Suominen
@jksuom
You could take a look at how Sylow subgroups are found in SymPy and then try to find an algorithm for Hall subgroups.
Mohit Gupta
@mohitacecode
Sure I will do it now.
Mohit Gupta
@mohitacecode
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)
Kalevi Suominen
@jksuom
Check if order of G is divisible by every prime in P
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.
Kalevi Suominen
@jksuom
Finding a Hall subgroup for several primes is probably harder than finding a Sylow subgroup.
Mohit Gupta
@mohitacecode

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.

can you explain please? beacuse to contain a subgroup for any prime the order of the group should be divisible by it sylow p subgroup also uses it.

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 .
Kalevi Suominen
@jksuom
The intersection of a Sylow 2-group and 3-group is a subgroup of both of them. The order of a subgroup of a 2-group is a power of 2 and that of a 3-group is a power of 3. The only subgroup whose order is both a power of 2 and a power of 3 is the trivial group of order 1.
Mohit Gupta
@mohitacecode
okay now I understood
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 of sylow 2 subgroup and sylow 3 subgroup for any group G.just clarifying .

I was completely wrong

Mohit Gupta
@mohitacecode
any subgroup of index 2 is normal is the condition which we can used in is_normal method in perm_groups.py which will reduce further unnecessary steps. please see If I am saying anything wrong
Kalevi Suominen
@jksuom
Computation of the index will also take time. Besides, index 2 is probably rather uncommon.
Mohit Gupta
@mohitacecode
computatation of index requires to compute degree maybe it is much efficient than going for computing further steps.
Mohit Gupta
@mohitacecode
Besides, index 2 is probably rather uncommon.
So should I leave it ?
Kalevi Suominen
@jksuom
Perhaps you could check how adding that test would affect the average time spent in in_normal.
Mohit Gupta
@mohitacecode
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
Mohit Gupta
@mohitacecode
• 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.
Mohit Gupta
@mohitacecode
Addition of method to compute 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)
Kalevi Suominen
@jksuom
I think that something like that has already been implemented.
Mohit Gupta
@mohitacecode
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")
Kalevi Suominen
@jksuom
Polycyclic groups are actually special fp-groups. They are formed from a free group by relators that are called pc_presentation.
Mohit Gupta
@mohitacecode
Okay understood will try to use it with polycyclic group
Mohit Gupta
@mohitacecode
Can you please tell how should I use this method with polycyclic group?