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In his article of 1993/01/15 Peter M"uller writes:

The list PGTable in the primitive groups library contains some useful

data about the primitive groups of degree \le 50.

In practice one often likes to know which class of the O'Nan Scott

classification a primitive group belongs to. Wouldn't it be sensible

to add this information as a further component? Additionally,

generators of one of the (at most two) minimal normal subgroups would

be fine, as GAP doesn't seem (or did I miss something?) to offer a

convenient way to compute them.

Several comments.

1) Sims says in his article "Computational methods in the study of

permutation groups" in "Computational Problems in Abstract Algebra"

(Proceedings Oxford 67, John Leech ed.), which contains the primitive

groups of degree <= 20:

Any primitive group G of degree n < 60 has a unique minimal

normal subgroup N.So how comes you talk about "the (at most two) minimal normal

subgroups"? Or am I misunderstanding something here?

2) If the primitive group G of degree p^k has an elementary abelian

regular normal subgroup, this is can be computed as

Core( G, SylowSubgroup( G, p ) )Peter Neumann in his article "Some algorithms for computing with

finite permutation groups" in "Proceedings of Groups - St. Andrews

1984" (E.F. Robertson & C.M. Campbell ed.) gives a better algorithm,

which avoids the computation of the Sylow subgroup. But for the

groups in GAP's primitive group library this simple approach seems to

be ok.

3) If the primitive group G of degree p^k < 5^5 is not of the affine

type the socle (i.e., because of 1) the minimal normal subgroup)

is the last term in the derived series, and can be computed as

d := DerivedSeries( g ); d[Length(d)];

4) The library file 'grp/primitiv.grp' identifies the minimal normal

subgroup, if it is again primitiv. For example for

g := PrimitiveGroup( 28, 11 );the corresponding line in 'grp/primitiv.grp' reads

[28, 29484, 2, 01000, 2805, 2707, "PZL(2,27)", 148,151,156,171],it tells us (in encoded form in the 5. column) that the minimal

normal subgroup is 'PrimitiveGroup( 28, 5 )', and thus we get is asAsSubgroup( g, PrimitiveGroup( 28, 5 ) );If the fifth entry reads 'EABL' the minimal normal subgroup is

elementary abelian, and thus we can apply 2). If the fifth entry

is empty the minimal normal subgroup is not elementary abelian and

not primitive, and thus we should apply 3).

Hope this helps, Martin.

-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, D 51 Aachen, Germany

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