[GAP Forum] Order of elements in a group and their statistics

[muhammad shah]

Dear Stefanos,
Imitating and modifying a function "orderFrequency" of  Alexander Hulpke,
your function of spectrum should be as follows.
spectrum:= function(g)
local h,w;
w:= [];
w:= h -> Set(Elements(h), Order);
return w(g);
end;
See also the following application of the function
gap> s3:=SymmetricGroup(IsPermGroup,3);
Sym( [ 1 .. 3 ] )
gap> s7:=SymmetricGroup(IsPermGroup,7);
Sym( [ 1 .. 7 ] )
gap> spectrum(s3);
[ 1, 2, 3 ]
gap> spectrum(s7);
[ 1, 2, 3, 4, 5, 6, 7, 10, 12 ]
Hopes this helps,
Muhammad Shah


> Date: Fri, 12 Jul 2013 22:58:55 +0100
> From: stefanosaivazidis at gmail.com
> To: forum at gap-system.org
> Subject: [GAP Forum] Order of elements in a group and their statistics
> 
> Dear forum,
> 
> I have the following (rather naive) question to ask: what is the
> most efficient way to find the spectrum of a finite group G, and
> compute for each integer in the spectrum the number of elements with
> given order? An integer d lies in the spectrum of G iff there exists
> at least one g in G such that o(g)=d.  The algorithm, I imagine, should
> proceed along these lines:
> 
> 1) define the group G,
> 2) compute the set of divisors of |G| and store this as a list L,
> 3) refine L (by excluding those divisors of |G| which do not
>      appear as element orders) to obtain the spectrum of G and
>      store this in a new list L',
> 4) compute how many elements of G have order d, for each d in L'
> 
> Your thoughts on how to make this precise algorithmically would
> be much appreciated. Also, is it possible to produce a graph with
> the statistics found by the main programme?
> 
> Many thanks in advance.
> 
> Best wishes,
> Stefanos
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