[GAP Forum] I am puzzled by the function DihedralGroup(n)!

[Ramon Esteban]

Dear colleague, dear forum,

* ?????? <dongjc at njau.edu.cn> [071127 04:41]:
> Dear Forum,Dear Everyone:
> As we all know, a dihedral group  $D_{2n}$ is generated by two elements $a$ and $b$ such that $a^n=1,b^2=1$ and $aba=b$. Hence $D_{2n}$ consists of  $2n$ elements. However the following statements puzzle me:
> gap>d8:=DihedralGroup(8);
> <pc group of size 8 with 3 generators>
> gap>d10:=DihedralGroup(10);
> <pc group of size 10 with 2 generators>
> The first statement implies that d8 has 3 generators. The second statement implies that d10 has 2 elements. WHY?

This only means that d8 has a generating system with 3 elements, but
it does not mean that the smallest generating system must have 3
elements. For instance, the elements d8.1, of order 2, and d8.2, of
order 4, generate the dihedral group:

gap> d8:=DihedralGroup(8);
<pc group of size 8 with 3 generators>
gap> Group(d8.1,d8.2)=d8;
true

This generating system corresponds to an internal representation of
the polycyclic (soluble) groups d8 and d10 in which GAP considers a
composition series and takes a representative in the group of an
element from each composition factor. Every composition series of d8
has 3 composition factors, while every composition series of d10 has 2
factors. This explains the number of generators you have found. You
can find the details about the representation of polycyclic groups in
the chapter "Pc Groups" of the manual:

gap> ?Pc Groups

Perhaps you are interested in the function MinimalGeneratingSet:

gap> MinimalGeneratingSet(d8);
[ f1, f2 ]
gap> last=[d8.1,d8.2];
true

Hope this helps.

Best wishes,
-- 
Ramon <resteban at mat.upv.es>

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