O'Nan-Scott theorem

A reduction theorem for the class of finite primitive permutation groups, distributing them in subclasses called types whose number and definition may vary slightly according to the criteria used and the order in which these are applied. Below, six types are described by characteristic properties, additional properties are given, a converse group-theoretical construction is presented, and a few small examples are given.

Let be a finite set and let be a primitive permutation group on . Then the stabilizer of a point belonging to is a maximal subgroup of containing no non-trivial normal subgroup of . Conversely, and constructively, this amounts to the data of a group and of a maximal subgroup containing no non-trivial normal subgroup of ; the elements of are the left cosets with in , and the action of on is by left translation.

The reduction is based on a minimal normal subgroup of . Either is unique or there are two such, each being regular on and centralizing the other (cf. also Centralizer). The socle, , of is the direct product of those two subgroups. The subgroup is a direct product of isomorphic copies of a simple group , hence with for and . One puts , . Fixing a point of , let be the orbit of under and let be the intersection of the , .

One of the criteria of the reduction is whether is Abelian or not (cf. Abelian group), and another is to distinguish the case from . Still another criterion is to distinguish the case where is regular or not. If is non-Abelian, then acts transitively on the set and it induces a permutation group on it with in the kernel of the action. The nature of provides another property. A final property is whether is reduced to or equal to . The affine type is characterized by the fact that is unique and Abelian. Then is endowed with a structure of an affine geometry whose points are the elements of , is a prime number and is the dimension, with . Thus and is a subgroup of the affine group containing the group of all translations. Also, the stabilizer of is an irreducible subgroup (cf. also Irreducible matrix group) of .

Conversely, for a finite vector space of dimension over the prime field of order and an irreducible subgroup of , the extension of by the translations provides a primitive permutation group of affine type.

Examples are the symmetric and alternating groups of degree less than or equal to four (cf. Symmetric group; Alternating group), and the groups where is a prime power.

The almost-simple type is characterized by , , and non-Abelian. It follows that is not regular and that ; namely, is isomorphic to an almost-simple group.

Conversely, the data of an almost-simple group and one of its maximal subgroups not containing its non-Abelian simple socle determines a primitive group of almost-simple type.

Examples are the symmetric and alternating groups of degree (cf. Symmetric group; Alternating group), the group acting on the projective subspaces of a fixed dimension, etc.

The holomorphic simple type is characterized by and the fact that there are two non-Abelian regular minimal normal subgroups. Moreover, , and is described as the set of mappings from onto of the form , where and varies in some subgroup of . Conversely, for any non-Abelian simple group the action on the set of elements of provided by the mappings , where and varies in some subgroup of , gives a primitive group of holomorphic simple type.

Examples occur for the degree with , for the degree with , etc.

The twisted wreath product type is characterized by the fact of being non-Abelian, being regular and unique. Then , . The stabilizer is isomorphic to some transitive group of degree whose point stabilizer has a composition factor isomorphic to . The smallest example has degree with .

A converse construction is not attempted here.

For the next descriptions of types some preliminary notation and terminology is needed.

Let be a set of cardinality and let be some integer. Consider the Cartesian product, or, better, the Cartesian geometry, which is the set equipped with the obvious Cartesian subspaces obtained by the requirement that some coordinates take constant values, and with the obvious Cartesian parallelism. Each class of parallels is a partition of . If is a point of , then there are Cartesian hyperplanes containing and each of the Cartesian subspaces containing corresponds to a unique subset of that set of hyperplanes. denotes the automorphism group. For a fixed coordinate () there is a subgroup of fixing each coordinate except , and is isomorphic to the symmetric group of degree . The direct product is the automorphism group mapping each Cartesian subspace to one of its parallels. Also, induces the symmetric group of degree on the set .

The product action of a wreath product type is characterized by , non-Abelian and . Then is primitive. Also, is intransitive, the set bears the structure of a Cartesian geometry invariant under and whose Cartesian hyperplanes are the and their transforms under , and is parallel to its transforms under . Each leaves each Cartesian line in some parallel class invariant. The group stabilizing a Cartesian line induces on it some primitive group with as minimal normal subgroup, which is a group of almost-simple type or of holomorphic simple type. The distinction between these two cases is characterized by being not regular or being regular, respectively.

Conversely, given a primitive group of almost-simple type or holomorphic simple type with minimal normal subgroup on the set and a primitive group of degree , these data provide a wreath product group with a product action on the Cartesian geometry , in which is a minimal normal subgroup of and the are the Cartesian hyperplanes of containing a given point.

Examples occur for , of cardinality five and equal to or ; also, for of cardinality six and one of or , etc.

The diagonal type is characterized by the fact , is non-Abelian, is not regular, and . Then is primitive. Also, each is transitive on and is semi-regular. Moreover, , and is regular for all . Let a "line" be any orbit of some . Call two lines "parallel" if they are orbits of the same . For each , the lines that are not orbits of constitute the Cartesian lines of a Cartesian space of dimension on . This geometric structure is called a diagonal space.

A converse construction is not given here. The smallest examples occur for and , hence for .

See also: Permutation group; Primitive group of permutations; Symmetric group; Simple group; Wreath product.

References

[a1]	M. Aschbacher, "The subgroup structure of the finite classical groups by Peter Kleidman and Martin Liebeck" Bull. Amer. Math. Soc. , 25 (1991) pp. 200–204
[a2]	F. Buekenhout, "On a theorem of O'Nan and Scott" Bull. Soc. Math. Belg. B , 40 (1988) pp. 1–9
[a3]	J.D. Dixon, B. Mortimer, "Permutation groups" , GTM , Springer (1996) MR1409812 Zbl 0951.20001