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 $ \Omega $ be a finite set and let $ G $ be a primitive permutation group on $ \Omega $. Then the stabilizer $ G _ {O} $ of a point $ O $ belonging to $ \Omega $ is a maximal subgroup of $ G $ containing no non-trivial normal subgroup of $ G $. Conversely, and constructively, this amounts to the data of a group $ G $ and of a maximal subgroup $ L $ containing no non-trivial normal subgroup of $ G $; the elements of $ \Omega $ are the left cosets $ gL $ with $ g $ in $ G $, and the action of $ G $ on $ \Omega $ is by left translation.

The reduction is based on a minimal normal subgroup $ N $ of $ G $. Either $ N $ is unique or there are two such, each being regular on $ \Omega $ and centralizing the other (cf. also Centralizer). The socle, $ { \mathop{\rm soc} } ( G ) $, of $ G $ is the direct product of those two subgroups. The subgroup $ N $ is a direct product of isomorphic copies of a simple group $ S $, hence $ N \simeq S _ {1} \times \dots \times S _ {k} $ with $ S _ {i} \simeq S $ for $ i = 1 \dots k $ and $ k \geq 1 $. One puts $ S _ {i} ^ {v} = S _ {1} \times \dots \times S _ {i - 1 } \times S _ {i + 1 } \times \dots \times S _ {k} $, $ i = 1 \dots k $. Fixing a point $ O $ of $ \Omega $, let $ h _ {i} $ be the orbit of $ O $ under $ S _ {i} ^ {v} $ and let $ \Omega ^ {v} $ be the intersection of the $ h _ {i} $, $ i = 1 \dots j $.

One of the criteria of the reduction is whether $ S $ is Abelian or not (cf. Abelian group), and another is to distinguish the case $ k = 1 $ from $ k > 1 $. Still another criterion is to distinguish the case where $ N $ is regular or not. If $ N $ is non-Abelian, then $ G $ acts transitively on the set $ \Sigma = \{ S _ {1} \dots S _ {k} \} $ and it induces a permutation group $ P $ on it with $ { \mathop{\rm soc} } ( G ) $ in the kernel of the action. The nature of $ P $ provides another property. A final property is whether $ \Omega ^ {v} $ is reduced to $ O $ or equal to $ \Omega $. The affine type is characterized by the fact that $ N $ is unique and Abelian. Then $ \Omega $ is endowed with a structure of an affine geometry $ AG ( d,p ) $ whose points are the elements of $ \Omega $, $ p $ is a prime number and $ d $ is the dimension, with $ d \geq 1 $. Thus $ | \Omega | = p ^ {d} $ and $ G $ is a subgroup of the affine group $ AGL ( d,p ) $ containing the group $ N $ of all translations. Also, the stabilizer $ G _ {O} $ of $ O $ is an irreducible subgroup (cf. also Irreducible matrix group) of $ GL ( d,p ) $.

Conversely, for a finite vector space of dimension $ d $ over the prime field of order $ p $ and an irreducible subgroup $ H $ of $ GL ( d,p ) $, the extension of $ H $ 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 $ AGL ( d,q ) $ where $ q $ is a prime power.

The almost-simple type is characterized by $ k = 1 $, $ { \mathop{\rm soc} } ( G ) = N $, and $ N $ non-Abelian. It follows that $ N $ is not regular and that $ S \leq G \leq { \mathop{\rm Aut} } ( S ) $; namely, $ G $ 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 $ \geq 5 $( cf. Symmetric group; Alternating group), the group $ PGL ( n,q ) $ acting on the projective subspaces of a fixed dimension, etc.

The holomorphic simple type is characterized by $ k = 1 $ and the fact that there are two non-Abelian regular minimal normal subgroups. Moreover, $ | \Omega | = | S | $, and $ G $ is described as the set of mappings from $ S $ onto $ S $ of the form $ g \rightarrow ag ^ {s} b $, where $ a,b \in S $ and $ s $ varies in some subgroup of $ { \mathop{\rm Aut} } ( S ) $. Conversely, for any non-Abelian simple group $ S $ the action on the set of elements of $ S $ provided by the mappings $ g \rightarrow ag ^ {s} b $, where $ a,b \in S $ and $ s $ varies in some subgroup of $ { \mathop{\rm Aut} } ( S ) $, gives a primitive group of holomorphic simple type.

Examples occur for the degree $ 60 $ with $ S = { \mathop{\rm Alt} } ( 5 ) $, for the degree $ 168 $ with $ S = PSL ( 3,2 ) $, etc.

The twisted wreath product type is characterized by the fact of $ S $ being non-Abelian, $ N $ being regular and unique. Then $ | \Omega | = | {S ^ {k} } | $, $ k \geq 6 $. The stabilizer $ G _ {O} $ is isomorphic to some transitive group of degree $ k $ whose point stabilizer has a composition factor isomorphic to $ S $. The smallest example has degree $ 60 ^ {6} $ with $ S \simeq { \mathop{\rm Alt} } ( t ) $.

A converse construction is not attempted here.

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

Let $ A $ be a set of cardinality $ a \geq 2 $ and let $ n \geq 2 $ be some integer. Consider the Cartesian product, or, better, the Cartesian geometry, which is the set $ \Omega = A ^ {n} $ 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 $ \Omega $. If $ O $ is a point of $ \Omega $, then there are $ n $ Cartesian hyperplanes containing $ O $ and each of the $ 2 ^ {n} $ Cartesian subspaces containing $ O $ corresponds to a unique subset of that set of hyperplanes. $ { \mathop{\rm Aut} } ( A ^ {n} ) $ denotes the automorphism group. For a fixed coordinate $ i $( $ 1 \leq i \leq n $) there is a subgroup $ S _ {i} $ of $ { \mathop{\rm Aut} } ( A ^ {n} ) $ fixing each coordinate except $ i $, and $ S _ {i} $ is isomorphic to the symmetric group of degree $ a $. The direct product $ S _ {1} \times \dots \times S _ {n} $ is the automorphism group mapping each Cartesian subspace to one of its parallels. Also, $ { \mathop{\rm Aut} } ( A ^ {n} ) $ induces the symmetric group of degree $ n $ on the set $ \{ S _ {1} \dots S _ {n} \} $.

The product action of a wreath product type is characterized by $ k > 1 $, $ S $ non-Abelian and $ \Omega ^ {v} = \{ O \} $. Then $ P $ is primitive. Also, $ S _ {i} $ is intransitive, the set $ \Omega $ bears the structure of a Cartesian geometry invariant under $ G $ and whose Cartesian hyperplanes are the $ h _ {i} $ and their transforms under $ G $, and $ h _ {i} $ is parallel to its transforms under $ S _ {i} ^ {v} $. Each $ S _ {i} $ leaves each Cartesian line in some parallel class invariant. The group stabilizing a Cartesian line induces on it some primitive group with $ S _ {i} $ 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 $ N $ being not regular or being regular, respectively.

Conversely, given a primitive group $ X $ of almost-simple type or holomorphic simple type with minimal normal subgroup $ S $ on the set $ A $ and a primitive group $ P $ of degree $ k > 1 $, these data provide a wreath product group $ G = ( X _ {1} \times \dots \times X _ {k} ) { \mathop{\rm wr} } P $ with a product action on the Cartesian geometry $ A ^ {k} = \Omega $, in which $ N = S _ {1} \times \dots \times S _ {k} $ is a minimal normal subgroup of $ G $ and the $ h _ {i} $ are the Cartesian hyperplanes of $ \Omega $ containing a given point.

Examples occur for $ k = 2 $, $ A $ of cardinality five and $ G $ equal to $ { \mathop{\rm Alt} } ( 5 ) $ or $ { \mathop{\rm Sym} } ( 5 ) $; also, for $ A $ of cardinality six and $ G $ one of $ { \mathop{\rm Alt} } ( 6 ) $ or $ { \mathop{\rm Sym} } ( 6 ) $, etc.

The diagonal type is characterized by the fact $ k > 1 $, $ S $ is non-Abelian, $ N $ is not regular, and $ \Omega ^ {v} = \Omega $. Then $ P $ is primitive. Also, each $ \Omega _ {i} ^ {v} $ is transitive on $ \Omega $ and $ S _ {i} $ is semi-regular. Moreover, $ | \Omega | = | S | ^ {k - 1 } $, $ N _ {O} = S _ {i} $ and $ S _ {i} ^ {v} $ is regular for all $ i = 1 \dots n $. Let a "line" be any orbit of some $ S _ {i} $. Call two lines "parallel" if they are orbits of the same $ S _ {i} $. For each $ i $, the lines that are not orbits of $ S _ {i} $ constitute the Cartesian lines of a Cartesian space of dimension $ k - 1 $ on $ \Omega $. This geometric structure is called a diagonal space.

A converse construction is not given here. The smallest examples occur for $ S = { \mathop{\rm Alt} } ( 5 ) $ and $ k = 3 $, hence for $ | \Omega | = 3,600 $.

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