Finite imprimitive rank $3$ affine groups -- I
Cai Heng Li, Luyi Liu, Hanyue Yi, Yan Zhou Zhu
TL;DR
This work classifies imprimitive rank-$3$ affine permutation groups in characteristic $p$ under the non-$p$-local point-stabilizer condition $O_p(G)=1$, showing such groups are exceedingly rare. Using the affine model $V{:}G$ with $V=\mathbb{F}_p^n$, the authors develop a framework around the parabolic subgroup $P[W]=Q{:}L$, where $Q\cong W\otimes (V/W)^*$, and employ cohomology $\mathrm{H}^i$ to control complements and orbit structure. A key step is the reliance on the classification of non-solvable transitive linear groups to bound possibilities, culminating in the identification that, up to conjugacy, only two exceptional subgroups $G_1,G_2\cong\mathrm{GL}_3(2)$ inside $\mathrm{GL}_4(2)$ occur; these realize the two sought-after examples $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup. The results provide a sharp, explicit classification contributing to the broader program of rank-$3$ affine group classification and informing related combinatorial and geometric structures.
Abstract
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.