Pair Generators for Split Metacyclic Groups in GL(n,p)

[aidevnn]

An implementation of pair generators within GL(n,p) for split metacyclic groups has been successfully tested for orders up to 256. It would be beneficial to add in the project references a proven generalization of this result from existing literature.

[aidevnn]

MS-Copilot suggested this post on arxiv, which is relevant to this topic.

Generators of split extensions of Abelian groups by cyclic groups, Luc Guyot

[aidevnn]

The generators in $\textbf{GL}(n,p)$ for the group $\textbf{M} \rtimes \textbf{C}_n$, where $\textbf{M}$ is Abelian and $\textbf{C}_n$ is cyclic, have been determined up to order 63 through exhaustive search. This search utilized diagonal matrix generators for $\textbf{M}$ and a permutation matrix generator for $\textbf{C}_n$.

Before proceeding with groups of order 64 and higher, reducing the space of possibilities would be interesting.

[aidevnn]

Once an effective heuristic is found for generators in $\textbf{GL}(n,p)$ for the group $\textbf{M} \rtimes \textbf{C}_n$, there will be no further need to name subgroups. Calling the backend methods used for naming subgroups, at

https://github.com/aidevnn/FastGoat/blob/3488668b0c263d5f17ac485a68568feed3c3b996/FastGoat/Structures/Subgroups/AllSubgroups.cs#L123-L132

will be more efficient. Subsequently, this code can be moved from the Program.cs file to the Examples namespace.

Instead of improving the current implementation at this moment, it may be more beneficial to address the environment issues while upgrading to Ubuntu 24.04 or tackle other group theory-related problems.

[aidevnn]

The subgroup of $\text{GL}(n,p)$ generated by the diagonal and permutation matrices is interesting to study. This subgroup is significant for matrix form of abelians, metacyclics and many others groups.

[aidevnn]

The matrix representation for groups of order up to 63 is nearly complete, except for one (C2 . S4 with GAP ID (48,28)). However, after considering metacyclic groups and the semidirect product of abelian and cyclic groups, finding a practical approach for solvable groups of order more than 64, appears more challenging.

According to Cayley's theorem, a group of order $n$ is always a subgroup of $\mathbf{S}_n$ and also of $\text{GL}(n,2)$. However, using the subgroup of $\text{GL}(n_1, p)$ generated by the diagonal and permutation matrices is interesting when $n_1 < n$, and, characters table in $\text{GL}(n_1,\mathbb{C})$ gives always useful information on $n_1$ and $p$.

[aidevnn]

The matrix forms of the groups SL(2,3) and GL(2,3) are not currently generated by diagonal or permutation matrices. In particular, A5 poses a significant challenge.

[aidevnn]

Groups generated by diagonal or permutation matrices in $\text{GL}(n,p)$ up to order 63 have been successfully completed. Additionally, they provide an easy and direct way to obtain representation in $\text{GL}(\mathbb{C})$, and the correctness can be verified using character tables.

Conversely, starting with the character table of a finite group and then programmatically finding an isomorphic group generated by diagonal or permutation matrices in $\text{GL}(n,p)$ with a better control over $n, p$ and the space of all possibilities, is an interesting goal to achieve.