[GAP Forum] Canonical form for some small groups and efficient characterisation of the generalized symmetric groups
Max Horn
max at quendi.de
Tue Dec 19 17:51:13 GMT 2017
Dear Martin,
> On 19 Dec 2017, at 16:43, Rubey Martin <martin.rubey at tuwien.ac.at> wrote:
>
> Dear Max,
>
>>>> The permutation action on the roots would be a rather natural choice.
>
>> Regarding Nicolas' suggestion: While it is a "natural" choice, it is not
>> necessarily a "canonical" choice. At the very least, you'd have to define
>> how to label the roots "canonically".
>
> let me ask: are you saying that that there are both "bad" but also
> "canonical labellings" of the roots? (let's consider only finite type)
>
> Or put differently: do you have an example at hand where different labellings
> of the roots yield non-conjugate permutation groups?
No, that's not what I meant. Of course any labeling with numbers 1..n can be
transformed into any other by a permutation, hence the resulting permutation
groups are conjugate in Sym(n).
Thus any choice of such a labeling determines a conjugacy class -- but
*only* that. It does not determine a representative of such a class.
The question then is: How useful is that? Since you are talking about a
database and finding things in it, I assumed you wanted a "canonical form"
which is a string, or perhaps a tuple of integers and strings, or something
"simple" like that. A conjugacy class of subgroups isn't that, though.
Cheers,
Max
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