sympy/GroupTheory

Mohit Gupta

@mohitacecode

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

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)

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?

Mohit Gupta

@mohitacecode

should I consider looking for monoids and semigroups as a one of usefull idea as it was also mentioned by gaurav(@gxyd ) in gsoc 2016 but was'nt completed due to time constarint.

Kalevi Suominen

@jksuom

Monoids are more difficult to handle than groups. I don't know of any algorithms for them.

Mohit Gupta

@mohitacecode

Okay thanks for you response I will consider these things if i add it in proposal.

Mohit Gupta

@mohitacecode

Till now socle of a primitive permutation group is not implemented which can be further extended in calculating a chief series for a permuation group.I am considering this for adding it in my proposal.

After doing some research I have find some implementation of pseudo code for them.

Kalevi Suominen

@jksuom

The socle is not implemented for any finite group. That would probably be a good addition but maybe not easy.

Mohit Gupta

@mohitacecode

Thanks for your valuable suggestion kalevi :)

Mohit Gupta

@mohitacecode

I looked at the algorithm describes to see the 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.

Mohit Gupta

@mohitacecode

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

Kalevi Suominen

@jksuom

I wonder how that would work with a permutation group of order >= 10^6.

Mohit Gupta

@mohitacecode

can we do something like if the order of group is greater than certain no. then we can go for the former algorithm

or something like if the group is primitive then we can go for former one because the later algorithm doesnt take any advantage of the group being primitive

Mohit Gupta

@mohitacecode

Good morning , kalevi
Currently conjugacy classes of permutation group are calculated using naive algorithm It can be improved by using random method presented here Pg-94 in book Groups ’93 Galway/St Andrews (https://ru.b-ok2.org/dl/703924/e2e2ff).

do implementation look feasible to you?

Kalevi Suominen

@jksuom

It is hard to say. There are very little details. Maybe the only way to find out is to implement it.

Mohit Gupta

@mohitacecode

Ohhkay... I will probable add it in my proposal.

Mohit Gupta

@mohitacecode

just one thing I wanna ask :
In pc_presentation(dictionary) which is created in the pc_relators method. since pc_presentation is a dictionary can any value associated with key be exponential something like {x:x**2}

Kalevi Suominen

@jksuom

I think that this should mean that x**2 is the identity.

Mohit Gupta

@mohitacecode

~~yes x**2 is the identity.~~

Mohit Gupta

@mohitacecode

Actually It is initialised using free_group.identity and after that it multiplied with the value of the perm_to_free (map pcgs gen to free_group gen) and then it is collected using collected_word. that is what x**2 is.

Divyanshu Thakur

@divyanshu132

can any value associated with key be exponential something like {x:x**2}

value mapped to a particular key can take any form but it should be collected and equivalent to lhs.

Mohit Gupta

@mohitacecode

okay got it..

Mohit Gupta

@mohitacecode

is there any way to find the representative of a conjugacy class?

Mohit Gupta

@mohitacecode

Hello everyone,
I have uploaded my GSoc 2020 Proposal here (https://docs.google.com/document/d/1qFXjinWnO1NBQAeuHm_-ARVnlGM7Lx2XzZ3n2AXtc0Y/edit?usp=sharing) please have a look your suggestions are really required here and please say if I miss out anything.

Thank you very much for your time.
Mohit Gupta

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