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.
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:
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.
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]
andb[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.
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 subgroup
andsylow 3 subgroup
for any group G.just clarifying .
I was completely wrong