Calculations in group algebras and Finite groups as quotients of finitely presented groups

[Bharathwaj Palvannan]

Hi, 

I am trying to work with group rings of the form Q[G], where Q is the field of rational numbers and G is a finite (not necessarily abelian) group. To create the group G, I have tried to express the group G as a quotient of a free group, using some of the techniques mentioned here.  To be concrete, I have created a public worksheet here, where I have tried to work with Q[G], where G is a non-abelian group of order 125.  The group G is expressed as a quotient of the free group on e, f, where the relations are {e^5, f^25, efe^{-1}f^{-6}}. (I believe this works since the output of the command G.order() is 125). 

I am ultimately interested in doing computations over group rings:  for eg, let's say I am trying to compute xy + zwu, where x,y,z,w,u are elements of the group ring.  It would really be helpful to obtain these computations in their simplest forms. However, when I try to do some computations, even some simple ones involving the group elements, the expressions don't simplify and I cannot figure out a way to simplify them automatically. For eg the element e^5 does not get simplified to 1. I would like to work with complicated expressions and it will really help to obtain some workaround. 

Any help would be great.

Thank you.