Solvable primitive permutation groups of low rank

对于一个作用在$V$上的极大可解本原置换群$G$, $G$的唯一极小正规子群$T$是阶为$p^f$的elementary abelian群，且$T$正则地作用在$V$上。因此$G$可以看作在向量空间${\mathbb{F}}_p^f$上的作用，$G=T:G_0$, $0$向量的点稳定子$G_0$是一个线性群，群的秩就对应于线性群 $G_0$ 在 $V$ 上的轨道数量。

可解线性群可以分成三类：

• A：仿射直线上的仿射变换群（collineation groups of affine lines）。
• B：向量空间本原的群（vector space primitive）。
• L：向量空间非本原的群（imprimitive）。

Class A

Assume that $G_0=H$ has a maximal normal abelian subgroup $A$. Lemma 2.2 say, if $A$ is irreducible, then $G=T:H$ is a subgroup of the affine group of dimension $1$ over $GF(p^f)$, i.e.

$$G\subseteq \{x\to a{x}^{p^b}+c:x,a,c\in GF(p^f),1\le b\le f\}.$$

Definition: Let $\mathfrak{A}$ be the class of primitive permutation groups which are subgroups of finite one-dimensional affine groups.

Now let $G=T:H$ be a solvable primitive permutation group which is not in $\mathfrak{A}$. Let $G\in \mathfrak{B}$ if $H$ is primitive on $V$, and let $G\in \mathfrak{L}$ if $H$ is imprimitive.

Class L

In this case, $A$ is not unique: Suppose $G\in \mathfrak{L}$. Let $V=V_1\oplus \cdots \oplus V_r(r>1)$ be a decomposition of $V$ into minimal imprimitivity components of $H$. Since $H$ is irreducible, $H$ acts transitively on the set $\left\{V_i\right\}(1\leqq i\leqq r)$. For $H$ in $\mathcal{E},A$ is not unique; in fact, $A$ depends on the decomposition of $V,V=V_1\oplus \cdots \oplus V_r$.

Assume that $H$ acting on $V_r$ has $n$ orbits. Since $\{V_1,\cdots ,V_r\}$ forms a block system $\mathcal{B}$, there is $H^{\mathcal{B}}⩽S_r$. Note that $v\in V$ can be written as $v=(v_1,\cdots ,v_r)$ with $v_i\in V_i$, and its orbit is determined by “orbits of $v_i$ in $H$ on $V_i$“. If $H^{\mathcal{B}}=S_r$, then $\mathrm{rank}(H)=\left(\genfrac{}{}{0ex}{}{n+r}{r}\right)$, which is a lower bound of $\mathrm{rank}(H)$ as in proposition 2.5.

Class B

In this case, a maximal normal abelian subgroup $A$ of $H$ is equal to $F^{\ast }$ and is unique: Let $G\in \mathfrak{B}$, and let $A$ decompose $V$ into $t$ equivalent irreducible subspaces $V_1,\dots$, $V_t$, where $t>1$ and $t\mid f$. Then $A$ acts faithfully on each $V_{\mathrm{i}}$ as scalar multiplication by elements of $F=GF\left(p^k\right)$, where $k=f/t$. Thus in particular, $A$ is cyclic and $|A|\mid p^k-1$. Moreover, $A$ is contained in no proper subfield of $GF\left(p^k\right)$. Therefore, each $V_t$ and hence $V$ is a vector space over $GF\left(p^k\right)$, under the action of $A$. It follows that ${\mathfrak{C}}_H(A)$ and $H={\Re }_H(A)$ operate on $V$ as groups of linear and semilinear transformations, respectively, over $GF\left(p^k\right)$; hence $\left[H:{\mathfrak{S}}_H(A)\right]$ divides $k$. Since $G$ is maximal, $A=F^{\ast }$. Moreover $A$ is unique. For if $A_1$ is another maximal normal abelian subgroup of $H$ which determines $F_1=GF\left(p^{k_1}\right)$, then $\left|A:A\cap A_1\right|\mid k_1$ and hence $|A|\mid k_1\left(p^{k_1}-1\right)$. Therefore, $k\leqq k_1$ by a number-theoretic lemma [8, 3.1], $[9,2.4]$. Similarly, $k_1\leqq k$, so $k=k_1$. Therefore, $\left(p^k-1\right)|k|A_1\cap A\mid$, so $A\cap A_1$ generates both $F$ and $F_1$. Thus $A=A_1$.

对于最复杂的 B 类群，其内部包含一个结构特殊的关键子群：极小非交换正规子群 N （minimal normal nonabelian subgroup N ）。这个 N 通常是一个 $q$-群，甚至是”额外特殊q－群”.

• 该方法最精妙之处在于：通过分析 N 中点的稳定子 $N_x$ 的性质。这些稳定子 $N_x$ 都是 N 的阿贝尔子群（abelian subgroups）。
• 一个关键的群论事实是：如果两个点在同一个$G_0$ 轨道中，那么它们在 N 中的稳定子 $N_x$ 和 $N_y$ 必须是共轭的。
• 因此，通过计算 N 中有多少个非同构的阿贝尔子群可以作为稳定子，就可以得到群的秩 r（G）的一个下界。

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TLDR

Solvable primitive permutation group can be seen as “linear group”, because it has the unique minimal normal subgroup, which is abelian and regular. Then the class of solvable primitive permutation group can be divided into three classes. These three classes are discussed separately.

1 Introduction

Motivation

B. Huppert has classified all finite solvable doubly transitive permutation groups. It is possible to generalize Huppert’s theorem to the class of two-dimensional solvable flag-transitive affine groups. However, a more natural generalization is the class of permutation groups of low rank. Huppert’s theorem is concerned with groups of rank $2$.

Goals & Methods

In this paper, we consider the finite solvable primitive permutation groups of rank $>2$. Using techniques similar to Huppert’s, it is possible to classify the maximal solvable primitive permutation groups of rank $3$, and to restrict the possibilities for rank $4$ groups to a small set.

The standard analysis of the class of solvable linear groups divides this class roughly into $3$ subclasses, $\mathfrak{U}$, $\mathfrak{B}$, and $\mathfrak{L}$. $\mathfrak{U}$ consists of collineation groups of affine lines; $\mathfrak{B}$ consists of the remaining groups which are vector space primitive, and $\mathfrak{L}$ of those which are imprimitive. It is explained here.

Huppert's theorem

Huppert’s theorem states that if $G$ is a solvable doubly transitive permutation group, then $G_0\in \mathfrak{U}$ with the exception of $13$ groups which are in $\mathfrak{B}$.

Main result

Now let $G$ be a maximal solvable primitive permutation group of degree $n$ on a set $S$. Then $G_0$ is a semilinear group on a vector space $V$ over a field $GF(p^k)$.

Theorem

Theorem

$r(G)\ge 5$ except possibly in the following cases:

2 Solvable primitive permutation groups

The standard analysis of maximal solvable linear groups is applied to primitive permutation groups. The groups of low rank in $\mathfrak{L}$ are discussed in this section, and the groups in $\mathfrak{B}$ are discussed in 3-9.

Let $G$ be a maximal solvable primitive permutation group of a finite set $V$. Let $T$ be a minimal normal abelian subgroup of $G$, so that $T$ is an elementary abelian group of order $p^f$, for some prime $p$ and integer $f$. Since $G$ is primitive, then $T$ is unique, $T$ is transitive on $V$, and $|T|=|V|=p^f$. Thus $G=T{G}_0$ is the split extension of $T$ by $G_0$. Moreover, it is possible to make $V$ into a vector space of dimension $f$ over $GF(p)$ by inducing the group addition of $T$ in $V$. Then $T$ and $G_0$ act as the group of translations of $V$, and as an irreducible group of linear transformations of $V$ over $GF(p)$ respectively.

Lemma

Suppose $G$ is a solvable primitive permutation group $G$ with $G_0=H$. Let $A$ be a maximal normal abelian subgroup of $H$. If $A$ is irreducible, then $G=TH$ is a subgroup of the affine group of dimension $1$ over $GF(p^f)$. I.e.,$$ G\subseteq{x\to ax^{p^b}+c:x,a,c\in GF(p^f),1\leq b\leq f}.

Let $\mathfrak{U}$ be the class of primitive permutation groups which are subgroups of finite one-dimensional affine groups. Now let $G=TH$ be a solvable primitive permutation group which is not in $\mathfrak{U}$. Let $G\in \mathfrak{B}$ if $H$ is primitive on $V$, and let $G\in \mathfrak{L}$ if $H$ is imprimitive.