New interface for G Module N

[aidevnn]

The new interface for manipulating G-Module N, when N is an abelian group, is cumbersome and not suitable for computation of extensions of N by G. It also can't be used for modeling character computations, but it only simplifies reading. Restricting computation for N abelian, or the center of N, becomes mandatory since it corrects many bugs occurred when computing extensions of Dihedral Group D8 by C2.

Perhaps it would be better to use the previous object for manipulating polynomials of multiple indeterminates over the finite field Fq. Reducing systems could be done with Groebner, which is already possible and tested for trivial action, and group action could be possible through permutations of unknowns but it needs to be tested for evidence.

The book, Cohomology of Finite Group from Alejandro Adem and R.James Milgram will be very useful for that. The subject is becoming challenging, profound, and captivating.

[aidevnn]

Decomposing the group N into Abelian elementary factors is a more effective approach for reducing systems obtained through cochains than decomposing into Abelian invariants.

The next improvement is how the systems of equations for 2Coboundaries and 2Cocycles can be used in conjunction to deduce 2Cohomology and compute only representatives of cosets, rather than exploring all possibilities.

[aidevnn]

It is now feasible to compute generators of H2(C2xC2xC2, C2xC2) or H2(C3xC3, C3xC3). Although the process is slow, it is significantly more efficient than a brute-force Depth-First Search (DFS).

However, the current version encounters difficulties computing generators of H2(G,N) when N contains Z/p^2Z subgroups.

[aidevnn]

Significant advancements have been made in computing H2(G, N) when the group G acts trivially on any abelian group N, and when G acts non-trivially on N cyclic pGroup, which is not always simple. However, the case when G acts non-trivially on N, a direct sum of pGroup, such as C4 x C8 or C3 x C9, remains to be addressed.

[aidevnn]

The newly implemented interface is operational and is better compared to the previous Depth-First Search (DFS) method. The next step in our development process will be to make it faster.