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    Divyanshu Thakur
    @divyanshu132
    That's right, Thanks!
    Mohit Gupta
    @mohitacecode
    Hello everyone, My name is Mohit Gupta I will directly come to point I want to work on Group Theory module in GSOC 2020.I have seen the idea page of Group Theory and saw the following things are unImplemented
    • Computation of various subgroups of infinite finitely presented groups
    • Computation of Galois groups for a given polynomial
    • Finding kernels of homomorphisms with infinite domains
    • Extend functionalities of polycyclic groups
    • Quotient groups
    • Automorphism groups
      I also have some Ideas that I want to add such as :
    • Triple Product Property (https://en.wikipedia.org/wiki/Triple_product_property) we can do it for both groups and sets.
    • Normal Groups - While learning about Triple Product Property I also found that currently we do not have Implemented Normal subgroups so maybe this will be a new addition.
    • Intersection of subgroups
      This is not the final ideas I have just briefed I am still researching and going through code base.(will be able to spend more time once my exams get overed.
      I am working in sympy from Jan 2020. Any suggestions are welcome. Please tell if any there are any stalled Pr or where I can start the work from .
      @jksuom @divyanshu132 @asmeurer @RavicharanN
    Kalevi Suominen
    @jksuom
    Have you also been thinking about possible implementations? I seems to me that there is time for only a few of those topics in one summer.
    Mohit Gupta
    @mohitacecode
    Actually these were the things that I have seen or noticed while working in sympy.
    Yeah you are right I think we have to select only few ideas out of them that can be possibly completed in a summer.I am still working to narrow down this list (I will come up with much more descriptive and sensible list asap my universities exam gets over).
    I was looking for your insight that what do you think about the ideas and what can be improved or removed to make a balanced amount of work for summer.
    Kalevi Suominen
    @jksuom
    About normal subgroups. Normality is a property. Either a subgroup is normal or it is not so there is no special implementation. What could be implemented is a normality test. I would look into GAP to see what was implemented there.
    Kalevi Suominen
    @jksuom
    • Computation of various subgroups of infinite finitely presented groups
      Can you elaborate on this?

    • Computation of Galois groups for a given polynomial
      This is a whole new world, theory of algebraic numbers, not much group theory.

    • Finding kernels of homomorphisms with infinite domains
      Do you have some idea of how?

    • Extend functionalities of polycyclic groups
      Can you elaborate?

    • Quotient groups
      Of what kind of groups?

    • Automorphism groups
      Of what kind of structures?

    • Triple Product Property (https://en.wikipedia.org/wiki/Triple_product_property) we can do it for both groups and set
      Is there something to implement?

    • Intersection of subgroups
      This could be useful though I don't know of algorithms.

    Mohit Gupta
    @mohitacecode

    Triple Product Property (https://en.wikipedia.org/wiki/Triple_product_property) we can do it for both groups and set.
    Is there something to implement?

    As Triple Product Property is a property we have to just implement test for this.I have seen the various algorithms for this property here (https://arxiv.org/pdf/1104.5097.pdf) to save your time you can directly head over to page no. 6 and 7 and page 8 to see the efficiency of each algorithms.

    I think to implement this it would not take much time except few problems I am having :

    I have thoughts like:

    • what should we do when groups are not normal ?
    • Does Normal group property really be efficient there?
    Kalevi Suominen
    @jksuom
    Normality is essential for the commutativity to hold.
    Mohit Gupta
    @mohitacecode

    Normality is essential for the commutativity to hold.

    yes we can say that.(every subgroup of abelian group is normal)

    Mohit Gupta
    @mohitacecode
    I will elaborate the ideas above soon.(I need little time)
    Kalevi Suominen
    @jksuom
    The application of the triple product property seems to be to multiplication of numerical matrices. I think that NumPy implements that kind of algorithms. SymPy's matrices are typically symbolic which means that the acceleration methods devised for numerical matrices are generally not useful in SymPy. Consider, for example, two nxn matrices with entries a[i, j] and b[i, j]. The entries of their product are sums of n products. That makes n^3 products altogether, and there is no way to reduce the exponent 3.
    Mohit Gupta
    @mohitacecode
    Normality is essential for the commutativity to hold.
    By this you mean that it is essential for Triple Product Property?
    Mohit Gupta
    @mohitacecode

    The application of the triple product property seems to be to multiplication of numerical matrices. I think that NumPy implements that kind of algorithms. SymPy's matrices are typically symbolic which means that the acceleration methods devised for numerical matrices are generally not useful in SymPy. Consider, for example, two nxn matrices with entries a[i, j] and b[i, j]. The entries of their product are sums of n products. That makes n^3 products altogether, and there is no way to reduce the exponent 3.

    Yes It exactly focus on fast matrix multiplication and you seems right too.

    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

    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 ?

    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.
    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.