Intersted to Collaborate on Group Theory

[AJ]

Hello everyone,

I'm Amit Jana. Currently I'm a first year PhD student in cryptology and security at Indian Statistical Institute, Kolkata. Before that, I did my B.Sc and M.Sc in Mathematics under Vidyasagar Univercity and also did M.Tech in computer science at Indian Statistical Institute, Kolkata. 

I'm very much interested to collaborate on algorithmic implementations of group theoretical problems that are mentioned in this project. For example, As we know, to find the kernels of a homomorphisms with infinite domains, one possible way may be to find the normal subgroup of the group(domain of the homomorphism) in an efficient way. Also to decide the isomorphisms of the groups, we have to check only two conditions as 1. checking the homomorphism property 2. Bijection (In another way if we can check efficiently that the kernel of this homomorphism is only the identity). I'm following your above mentioned book "Handbook of Computational Group Theory".

But I don't understand how to start/follow the GAP library to learn about the previous works. Would you please recommend me?

[Valeriia Gladkova]

Hi Amit,

what trouble are you having with the GAP library exactly? Their reference manual is here: https://www.gap-system.org/Manuals/doc/ref/chap0.html and the main website has a section on how to install it.

Checking if a given homomorphism is in fact an isomorphism is already possible (though determining if a general map is a homomorphism would probably be tricky in some cases). The task now is to determine if there exists an isomorphism between two given groups and, if so, try to find one. This is difficult in general. The handbook talks a bit about random isomorphism testing in chapter 11 I think.

As for kernels of homomorphism with infinite domains, do you have any ideas on computing normal subgroups of infinite groups? If we had an efficient algorithm for finding normal subgroups, we could determine if a given normal subgroup is part of the kernel but checking if its generators are in the kernel, but this still doesn't tell us if this normal subgroup is the whole of the kernel. Also, some subgroups may have infinitely many generators.