> < ^ Date: Tue, 21 Aug 2001 11:14:04 +0200 (MET DST)
> < ^ From: Bettina Eick <beick@tu-bs.de >
> < ^ Subject: Re: SmallGroups library in GAP

Dear Gordon, dear GAP-Forum,

- is the order in which the groups appear arbitrary or are
there some principles underlying it? I notice that the
abelian groups don't seem to appear as the first few groups
of each order; this is mildly surprising for lists of groups,
but maybe there is a deeper underlying principle of which I
am not aware.

There are some principles. They are induced by the algorithms used
to construct the groups. Since we employed four different construction
methods, the ordering principles for the groups may depend on the
considered group order.

Most of the groups have been constructed using the p-group generation
algorithm or the Frattini extension method. In these two cases the
first sorting criteria is the order of the socle of the Frattini factor
of the given group. For example, for the groups of order 6 we obtain
that both groups are equal to their Frattini factors and for their socles
we find that |Soc( S_3 )| = 3 and |Soc( C_2 x C_3 )| = 6. Thus the
non-abelian group S_3 is listed first.

Further sorting principles and an overview on the construction and
setup of the Small Groups library can be found in 'A millenium project:
constructing small groups' by Hans Ulrich Besche, myself and Eamonn
also be found on my web page http://www.mathematik.uni-kassel.de/~eick.

- is this order guaranteed to remain unchanged (so that if I
refer in a paper to "the 5044th group of order 512" then this
can definitely be recovered by the future reader)

Yes, the ordering of the groups contained in the Small Groups library
is guaranteed to remain unchanged! The authors of the Small Groups library
all believe that this is a very important feature of this library and
hence we are not going to change the ordering of the groups.

```Best wishes,
Bettina
```