[GAP Forum] Canonical form for some small groups and efficient characterisation of the generalized symmetric groups

[Martin Rubey]

Dear Josh, Dear Forum,

Joshua Hunt <joshuahunt at math.ku.dk> writes:

> I'm afraid I don't quite understand your objection to the SmallGroup
> library. What requirements does it not fulfil?

Essentially: it doesn't cover S_7, S_5 § Z_2, etc.

So: the small group ID is just perfect for the "small" objects, but it
simply doesn't work for enough groups obtained as images of maps from,
say, graphs to groups.

In fact, that's what you write here:

> I would also point out that "groups of order less than 47" doesn't
> cover a lot of groups you probably would care about
> (e.g. symmetric/alternating groups S_5/A_5 and bigger, as mentioned
> in your example).

Let me try to put it in different words:

when initialising the findstat database,

* first we compute the canonical forms of the small objects (for groups:
  order less than 47)
  
* then we apply the maps in the database (eg, "automorphism group of a
  graph", "Weyl group of a finite Cartan type", "center of a group"),
  and compositions of two and three maps, to all small objects (eg,
  graphs, finite Cartan types, groups,...), and compute canonical forms
  of those.

Does this make it clearer?

Best,

Martin