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. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
