View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100501.png" /> be a finite set and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100502.png" /> be a primitive [[Permutation group|permutation group]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100503.png" />. Then the [[Stabilizer|stabilizer]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100504.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100505.png" /> belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100506.png" /> is a maximal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100507.png" /> containing no non-trivial [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100508.png" />. Conversely, and constructively, this amounts to the data of a [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o1100509.png" /> and of a maximal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005010.png" /> containing no non-trivial normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005011.png" />; the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005012.png" /> are the left cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005013.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005015.png" />, and the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005017.png" /> is by left translation.

The reduction is based on a minimal normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005019.png" />. Either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005020.png" /> is unique or there are two such, each being regular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005021.png" /> and centralizing the other (cf. also [[Centralizer|Centralizer]]). The socle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005022.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005023.png" /> is the direct product of those two subgroups. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005024.png" /> is a direct product of isomorphic copies of a simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005025.png" />, hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005027.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005029.png" />. One puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005031.png" />. Fixing a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005033.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005034.png" /> be the orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005035.png" /> under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005036.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005037.png" /> be the intersection of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005039.png" />.

One of the criteria of the reduction is whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005040.png" /> is Abelian or not (cf. [[Abelian group|Abelian group]]), and another is to distinguish the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005041.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005042.png" />. Still another criterion is to distinguish the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005043.png" /> is regular or not. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005044.png" /> is non-Abelian, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005045.png" /> acts transitively on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005046.png" /> and it induces a permutation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005047.png" /> on it with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005048.png" /> in the kernel of the action. The nature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005049.png" /> provides another property. A final property is whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005050.png" /> is reduced to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005051.png" /> or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005052.png" />. The affine type is characterized by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005053.png" /> is unique and Abelian. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005054.png" /> is endowed with a structure of an [[Affine geometry|affine geometry]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005055.png" /> whose points are the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005057.png" /> is a prime number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005058.png" /> is the dimension, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005059.png" />. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005061.png" /> is a subgroup of the affine group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005062.png" /> containing the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005063.png" /> of all translations. Also, the stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005065.png" /> is an irreducible subgroup (cf. also [[Irreducible matrix group|Irreducible matrix group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005066.png" />.

Conversely, for a finite vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005067.png" /> over the prime field of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005068.png" /> and an irreducible subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005070.png" />, the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005071.png" /> 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|Symmetric group]]; [[Alternating group|Alternating group]]), and the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005072.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005073.png" /> is a prime power.

The almost-simple type is characterized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005075.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005076.png" /> non-Abelian. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005077.png" /> is not regular and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005078.png" />; namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005079.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005080.png" /> (cf. [[Symmetric group|Symmetric group]]; [[Alternating group|Alternating group]]), the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005081.png" /> acting on the projective subspaces of a fixed dimension, etc.

The holomorphic simple type is characterized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005082.png" /> and the fact that there are two non-Abelian regular minimal normal subgroups. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005083.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005084.png" /> is described as the set of mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005085.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005086.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005089.png" /> varies in some subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005090.png" />. Conversely, for any non-Abelian simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005091.png" /> the action on the set of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005092.png" /> provided by the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005093.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005095.png" /> varies in some subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005096.png" />, gives a primitive group of holomorphic simple type.

Examples occur for the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005097.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005098.png" />, for the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o11005099.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050100.png" />, etc.

The twisted wreath product type is characterized by the fact of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050101.png" /> being non-Abelian, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050102.png" /> being regular and unique. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050104.png" />. The stabilizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050105.png" /> is isomorphic to some transitive group of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050106.png" /> whose point stabilizer has a composition factor isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050107.png" />. The smallest example has degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050108.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050109.png" />.

A converse construction is not attempted here.

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050110.png" /> be a set of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050111.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050112.png" /> be some integer. Consider the [[Cartesian product|Cartesian product]], or, better, the Cartesian geometry, which is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050113.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050114.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050115.png" /> is a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050116.png" />, then there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050117.png" /> Cartesian hyperplanes containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050118.png" /> and each of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050119.png" /> Cartesian subspaces containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050120.png" /> corresponds to a unique subset of that set of hyperplanes. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050121.png" /> denotes the automorphism group. For a fixed coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050122.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050123.png" />) there is a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050124.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050125.png" /> fixing each coordinate except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050126.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050127.png" /> is isomorphic to the symmetric group of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050128.png" />. The direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050129.png" /> is the automorphism group mapping each Cartesian subspace to one of its parallels. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050130.png" /> induces the symmetric group of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050131.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050132.png" />.

The product action of a wreath product type is characterized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050134.png" /> non-Abelian and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050135.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050136.png" /> is primitive. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050137.png" /> is intransitive, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050138.png" /> bears the structure of a Cartesian geometry invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050139.png" /> and whose Cartesian hyperplanes are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050140.png" /> and their transforms under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050141.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050142.png" /> is parallel to its transforms under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050143.png" />. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050144.png" /> leaves each Cartesian line in some parallel class invariant. The group stabilizing a Cartesian line induces on it some primitive group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050145.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050146.png" /> being not regular or being regular, respectively.

Conversely, given a primitive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050147.png" /> of almost-simple type or holomorphic simple type with minimal normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050148.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050149.png" /> and a primitive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050150.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050151.png" />, these data provide a wreath product group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050152.png" /> with a product action on the Cartesian geometry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050153.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050154.png" /> is a minimal normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050155.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050156.png" /> are the Cartesian hyperplanes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050157.png" /> containing a given point.

Examples occur for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050159.png" /> of cardinality five and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050160.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050161.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050162.png" />; also, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050163.png" /> of cardinality six and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050164.png" /> one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050165.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050166.png" />, etc.

The diagonal type is characterized by the fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050167.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050168.png" /> is non-Abelian, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050169.png" /> is not regular, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050170.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050171.png" /> is primitive. Also, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050172.png" /> is transitive on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050173.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050174.png" /> is semi-regular. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050175.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050176.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050177.png" /> is regular for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050178.png" />. Let a  "line"  be any orbit of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050179.png" />. Call two lines  "parallel"  if they are orbits of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050180.png" />. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050181.png" />, the lines that are not orbits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050182.png" /> constitute the Cartesian lines of a Cartesian space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050183.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050184.png" />. This geometric structure is called a diagonal space.

A converse construction is not given here. The smallest examples occur for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050185.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050186.png" />, hence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110050/o110050187.png" />.

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Buekenhout,   "On a theorem of O'Nan and Scott"  ''Bull. Soc. Math. Belg. B'' , '''40'''  (1988)  pp. 1–9</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.D. Dixon,   B. Mortimer,   "Permutation groups" , ''GTM'' , Springer  (1996)</TD></TR></table>