[GAP Forum] O'Nan--Scott type

[Peter Cameron]

Dear Forum,

I have a question about the O'Nan--Scott type of a primitive group; I would
appreciate advice. I hope the forum i s the right place to ask this.

The GAP function ONanScottType classifies primitive permutation groups into
eight types, of which the first is "affine".

My view of the O'Nan--Sccott theorem is a bit different. It is, first, a
reduction for primitive groups, rather like the reduction from transitive to
primitive. Call a primitive group G on Omega "non-basic" if it preserves a
Cartesian product structure on Omega, i.e. an identification of Omega with
A^B for some sets A,B such that G is identified with a subgroup of
Sym(A) wr Sym(B) with the product action. Now a non-basic group is of wreath
product or twisted wreath product type, and a basic group of affine, diagonal
or almost simple type.

Except for a small problem.

For example, the group PrimitiveGroup(25,5), the wreath product of D(10) with
Sym(2), has ONanScottType(PrimitiveGroup(25,5))="1" (i.e. affine), even
though this group is not basic.

For applications to synchronization and other things, I need to be able to
identify the non-basic groups, and ONanScottType won't do this for affine
groups. I would like a function IsBasic, similar to IsPrimitive.

It seems to me that there are several possibilities:

1. Perhaps this problem is already solved by some GAP code, or can be easily
solved by existing technology.

2. Maybe I could take some of the code for ONanScottType and adapt it.

3. If G is affine then it has the form p^n:H, where H is an irreducible linear
group; G is basic if and only if H is primitive (i.e. not in Aschbacher class
2). I could construct the linear group and use matrix group code to test this.

4. Maybe I should just write a bare-hands program to do this.

Any thoughts?

Peter.