gap-system
Automorphisms of SL
For $G = SL(d,q) $, one can decompose any automorphism $\alpha : G \to G$ as
$$ \alpha = idgf $$
where $i$ is an inner automorphism of G, $d$ is a diagonal automorphism of G, $g$ is the graph automorphism and $f$ is a field automorphism. These are defined as
Diagonal Automorphism :
$d : G \to G$ where $G = SL(n,q)$ is a diagonal automorphism iff
$$ \exists D \in GL(n,q) \text{ s.t. } D_{ij} = 0 \text{ if } i \ne j \ \text{ and } \ d(U) = D U D^{-1} \ \forall U \in G $$
Field Automorphism :
$f :G \to G$ is a field automorphism iff there exists an automorphism $\sigma$ over $GF(q)$ such that $f(U)$ is just $U$ with every entry replaced by its image under $\sigma$ for all $U \in G$ .
Graph Automorphism :
The graph automorphism $g : G \to G$ is given as
$$ g(U) = (U^T)^{-1} \forall U \in G $$
I have added functions related to these definitions in this PR.
Hi,
I have a few questions (I'd go through these in order, there is no point answering '3', if it turns out '1' means we don't really want the function in core GAP!)
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Are you sure these functions are of general interest? In general we don't want to add lots of functions to core GAP which might not be of general interest, which could instead go in a package. Are these part of a project you are working on? If so, you could consider making a package out of the functions you are working on. If you think they may be of general interest to all GAP users I'm happy to discuss that of course.
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SLAutomorphismDecomposition looks like it's going to take a VERY long time, you are iterating over a number of groups, so this function could well take an extremely long time to run.
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There are no tests
@pranav-joshi-iitgn what are your intentions with this PR? Do you wish to update it? Or shall we close it?
@pranav-joshi-iitgn what are your intentions with this PR? Do you wish to update it? Or shall we close it?
Since the final code didn't actually use any of this, I don't intend to update it. It'll be fine if you close it.