[GAP Forum] Exponent of Automorphism groups of small groups

[Alexander Konovalov]

Dear Yassine,

On 30 Mar 2014, at 12:31, yassine Guerboussa <yassine_guer at hotmail.fr> wrote:

> Dear Forum,
> I'm trying to compute the exponent of the automorphism groups of some small groups.For instance consider:
> G :=SmallGroup(5^6,11);SetInfoLevel(InfoAutGrp, 3);A := AutomorphismGroup(G); (the calculation is done without a problem)
> The command "Exponent(G);" gives the exponent of G without a problem, However  ("Exponent(A)") does not work (Perhaps because A is very large).
> Is there a way to compute the exponent of A, and (directly) the exponent of a p-Sylow in A?
> Thanks in advance.
> Yassine Guerboussa

In GAP 4.7.4, this calculation is easily achievable - it takes ~17 seconds on my machine:

gap> G :=SmallGroup(5^6,11);SetInfoLevel(InfoAutGrp, 3);A := AutomorphismGroup(G);    
<pc group of size 15625 with 6 generators>
#I  step 1: 5^6 -- init automorphisms 
#I    init automorphism group : Full 
#I    nice init group
#I  final step: convert
<group of size 11064475422000000000000000 with 4 generators>
gap> Exponent(A);time;
7931523600
16828

Of course, you can also compute Sylow p-subgroups and get their exponent too.

This is a result of an improvement introduced in GAP 4.7.2, see http://www.gap-system.org/Manuals/doc/changes/chap2.html, namely: "Computing the exponent of a finite group G could be extremely slow. This was due to a slow default method being used, which computed all conjugacy classes of elements in order to compute the exponent. We now instead compute Sylow subgroups P_1, ..., P_k of G and use the easily verified equality exp(G) = exp(P_1) x ... x exp(P_k). This is usually at least as fast and in many cases orders of magnitude faster.”

If you are using GAP 4.6 or older version, please upgrade to the latest GAP 4.7.4 release.

Best regards
Alexander