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

[Vee Kay]

On 12/07/13 22:58, Stefanos Aivazidis wrote:
> 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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Are steps 2 and 3 necessary then? I would imagine something like,

i) define the group G
ii) for each element in G, compute its order. add it to the spectrum (if 
not already there) and increment by one the number of times it appears 
(in a separate list presumably).

Don't know if there'd be a more efficient way to do this.

VK