List of Affine 2-transitive Groups

Note that if $X=V:G⩽{\mathrm{AGL}}_d(p)$ is affine, where $V={\mathbb{Z}}_p^d$ is elementary abelian, then $X$ is 2-transitive if and only if $G⩽{\mathrm{GL}}_d(p)$ is a transitive linear group (that is, $G$ acts transitively on the non-zero vectors of ${\mathbb{F}}_p^d$ ).

Proposition

There are four infinite classes of finite transitive linear groups.

• $G⩽\Gamma L\left(1,p^d\right)$;
• $G▹SL(a,q)$ and $p^d=q^a$;
• $G▹Sp(2a,q)$ and $p^d=q^{2a}$;
• $G▹G_2(q{)}^{\prime },p^d=q^6$ and $p=2$.

Sporadic finite transitive linear groups are listed in the following table.

Remark that the value for primitive identification is incorrect.

Cameron gives the number of actions of these $2$-transitive affine groups. Remark that $3^4:\mathrm{SL}(2,5)$ is missed. By magma, number of actions of $\mathrm{SL}(2,5)$ on $3^4$ is $1$.

G:=SL(2,5);
Irrs:=IrreducibleModules(G, GF(3));

M1:=Irrs[2];
M2:=Irrs[3];

Kernel(M1);
IsAbsolutelyIrreducible(M1);
CM1:=CohomologyModule(G,M1);
CohomologyGroup(CM1,1);

Kernel(M2);
IsAbsolutelyIrreducible(M2);
CM2:=CohomologyModule(G,M2);
CohomologyGroup(CM2,1);

MatrixGroup(2, GF(5)) of order 1
false
Full Vector space of degree 0 over GF(3)
MatrixGroup(2, GF(5)) of order 2
Generators:
    [4 0]
    [0 4]
true
Full Vector space of degree 1 over GF(3)

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The following theorem is used in Three Orbits Linear Groups.

Theorem

Let $G⩽\mathrm{GL}(V)$ be a transitive linear group acting on $V$ with $V\cong {\mathbb{F}}_p^n$. Then one of the followings holds.

• $G$ is solvable and either $G⩽{\mathrm{\Gamma L}}_1(p^n)$, or $G$ has a normal subgroup isomorphic to $2_{-}^{1+4}$ or ${\mathrm{SL}}_2(3)$, and $(n,p)\in \{(2,3),(2,5),(2,7),(2,11),(2,23),(4,3)\}$.
• $G^{(\infty )}$ is isomorphic to one of ${\mathrm{SL}}_m(q)$, ${\mathrm{Sp}}_m(q)$ and $G_2(q)$ ($m=6$ and $q$ even) with $q=p^f$ and $n=mf$. In this case, we say that $G$ is of classical type.
• $G^{(\infty )}$ is isomorphic to one of groups in $\{{\mathrm{SL}}_2(5),A_6,A_7,{\mathrm{PSU}}_3(3),{\mathrm{SL}}_2(13),2_{-}^{1+4}.A_5\}$, and $(n,p)\in \{(2,11),(2,19),(2,29),(2,59),(4,2),(4,3),(6,2),(6,3)\}$. In this case, we say that $G$ is of sporadic type. Also see here.

Here is a detailed table.

Proposition

Let $X=V:G$ be a $2$-transitive affine group. Then $V$ is the unique minimal normal subgroup.

\begin{proof} Let $N$ be a minimal normal subgroup of $X$. Since $X$ acting on $V$ is primitive, $N⊲X$ is transitive on $V$ by Orbits of Normal Subgroups Form a Block System. Then $|N|⩾|V|$. Since $V⊲X$ is a normal subgroup with order $|V|$, the subgroup $V$ is a minimal normal subgroup. If there exists another minimal subgroup $N$, then $N\cap V=\{1\}$ and so $N⩽C_X(V)$. Note that $C_X(V)=V$, so $N$ does not exist. Therefore, $V$ is the unique minimal normal subgroup. \end{proof}