a Summary for Rank 3 Groups

Classification of primitive rank three groups

Firstly, primitive permutation groups are classified. More details are written here: The O’Nan-Scott Theorem for Finite Primitive Permutation Groups, and Finite Representability.

Let $G$ be a nontrivial finite primitive permutation group on $\Omega$. Then $G$ is permutation isomorphic to a group that is either of affine type, twisted wreath type, almost simple type, diagonal type, or product type.

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By this theorem, primitive rank three groups can also be roughly classified.

Note that almost simple type is equivalent to whose socle is simple. So the groups under (iii) is of almost simple type. Also, (i) is of product type and (ii) is of affine type.

Note that a group of twisted wreath type or diagonal type is not rank three: here and here.

Theorem

If $G$ is primitive rank three group of finite degree $n$ then one of the following holds:

• $T\times T⊲G\le T_0\wr Z_2$, where $T_0$ is a $2$-transitive group of degree $n_0$, the socle $T$ of $T_0$ is simple and $n=n_0^2$;
• $G$ is an affine group, that is, the socle of $G$ is a vector space $V$, where $V\cong (Z_p{)}^d$ for some prime $p$ and $n=p^d$; moreover, if $G_0$ is the stabilizer of the zero vector in $V$ then $G=V{G}_0$, $G_0$ is an irreducible subgroup of ${\mathrm{GL}}_d(p)$, and $G_0$ has exactly two orbits on the non-zero vectors of $V$.
• the socle $L$ of $G$ is simple.

The groups under (i) can be determined using the classification of 2-transitive groups with simple socle.

A complete list of the rank $3$ groups under (ii) appears in The Affine Permutation Groups of Rank Three. In fact, Liebeck only considered the case of $G_0$ nonsolvable. For the case of $G_0$ solvable, note that all solvable primitive permutation group is of affine type, and Foulser finish the classification in Solvable primitive permutation groups of low rank.

Those under (iii) are determined

• in Kantor_Liebler_1982_The Rank 3 Permutation Representations of the Finite Classical Groups.pdf when the socle $\mathrm{soc}G$ is a classical group,
• in Maximal subgroups of low rank of finite symmetric and alternating groups when $\mathrm{soc}G$ is an alternating group,
• and in The Finite Primitive Permutation Groups of Rank Three when $\mathrm{soc}G$ is an exceptional group of Lie type or a sporadic group.

Classification of imprimitive rank three groups

As baykalovRankThreeInnately2023 said, imprimitive rank $3$ groups are so ‘wild’, imprimitive rank three groups are not completely classified up to now.

Two ways are tried:

By embedding theorem

By embedding theorem, a imprimitive rank $3$ group can be seen as a subgroup of a wreath product of two $2$-transitive groups.

Transclude of On-imprimitive-rank-3-permutation-groups#^him0rv

By Imprimitive Rank 3 Group is a Wreath Product of two 2-Transitive Groups, we have both $G_B^B$ and $G^{\mathcal{B}}$ are $2$-transitive groups.

Then by Burnside 2-Transitive, a $2$-transitive group has the unique minimal normal subgroup. Consider the socle $T$ of the component of $G$, that is, $G_B^B$. By Uniqueness of Block System of Imprimitive Rank 3 Group, $T$ is well-defined.

• If $T$ is nonabelian simple and $G$ is block-faithful, there is a rough classification: ^xhngii.
  • Almost simple block-faithful imprimitive rank $3$ groups has been classified: ^9182ab
• If $T$ is elementary abelian, that is, $H$ is a affine group:
  • On imprimitive rank 3 permutation groups gives a discussion in section 4.1.
  • [[Finite permutation groups of rank $3$#^35d38a|^35d38a]]

Smaller family

• Consider smaller family of imprimitive groups, like quasiprimitive, innately transitive and semiprimitive.
  • Quasiprimitive: ^502109
  • Innately transitive: baykalovRankThreeInnately2023
  • Semiprimitive: [[Finite permutation groups of rank $3$]]

Applications

Partial linear spaces

Partial linear spaces is defined as

A partial linear space which is not a graph or a linear space is called proper.

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Recall $2$-transitive graphs (they are trivial) and [[A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type#^78vy7v|rank $3$ graphs]] are classified directly by the classification of $2$-transitive groups and primitive rank $3$ groups, and $2$-transitive linear spaces are classified by Kantor, see here. Remark that $2$-transitive linear space on points $⟺$ $2$-ultrahomogeneous linear space, as any two points are collinear. So it remains to consider proper partial linear spaces.

As a preliminary, some notations are defined here:

Definitions of well-known families of partial linear spaces

Admitting primitive rank $3$ automorphism group

Consider the set of partial linear spaces which admit rank $3$ automorphism groups. The following three papers complete the classification of these partial linear spaces.

• Almost simple type: A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type
• Grid type: A classification of finite partial linear spaces with a primitive rank 3 automorphism group of grid type
• Affine type: Partial linear spaces with a rank 3 affine primitive group of automorphisms

Rmk. Note that Devillers only considers the case that $\mathrm{Aut}(\mathcal{S})$ is a primitive rank $3$ group, while BDFP consider its generalization, that is, $\mathrm{Aut}(\mathcal{S})$ has a subgroup $G$ which is primitive rank $3$. Since the latter case contains former, do not feel confused them.

Since most of partial linear spaces make no sense, they have little symmetry, which can be described by their automorphism groups. So we only want to figure out these highly symmetric partial linear spaces, like $2$-transitive partial linear spaces and flag-transitive partial linear space. As a generalization, we turn to consider 2-ultrahomogeneity partial linear space.

Motivation

As a generalization of $2$-transitive partial linear space, consider those partial linear spaces for which some automorphism group acts transitively on ordered pairs of distinct collinear points, as well as ordered pairs of distinct non-collinear points. It is defined by 2-ultrahomogeneity. Note that such partial linear spaces are flag-transitive. Furthermore, if they have non-empty line sets and are not linear spaces, then their automorphism groups are of rank $3$ on point set.

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Symmetric designs

Similarly, $2$-transitive symmetric designs are classified by Classification of 2-transitive symmetric designs-Kantor. As a generalization, we consider the design which has a automorphism subgroup $G$ and $G$ is primitive rank $3$. Then we can classify these designs:

• Almost simple and grid type: dempwolffPrimitiveRankGroups2001

• Affine type: W. Wirth? I doesn’t find reference.

  • in fact, Dempwolff finished the affine type in 2004, Affine Rank 3 Groups on Symmetric Designs.

• Here is a question: symmetric design is a special case of partial linear space, so it should be contained in last part? however, Devillers has some designs, which is less than Dempwolff.

Transitive decomposition

We gave a characterization of transitive decompositions:

• imprimitive, almost simple type: Rank 3 Transitive Decompositions of Complete Multipartite Graphs
• primitive, grid type: Transitive decompositions of graph products - rank 3 grid type

Recall:

Transclude of 2.0-for-winter-holiday#^rd1lbz

2-closures

On 2-closures of rank 3 groups