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Transitive decompositions of graph products: Rank 3 grid type. (English)
J. Group Theory 11, No. 2, 185-228 (2008).
A transitive decomposition is a pair $(Γ,{\cal P})$ where $Γ$ is a graph and ${\cal P}$ is a partition of of the arc set of $Γ$ such that there exist a group of automorphisms of $Γ$ which leaves ${\cal P}$ invariant and transitively acts on each part of ${\cal P}$. This paper concerns transitive decomposition where the group is a primitive rank 3 group of grid type. The graph $Γ$ in this case is the square grid or is the complement graph for the grid. It is proved (except in a small number of cases) that all transitive decompositions with respect to a rank 3 group of grid type can be characterized using generic constructions (Theorem 1.0.1). Theorem 5.0.8. Let $(Γ,{\cal P})$ be a $G$-transitive decomposition where $|{\cal P}|\ge 2$ and $G$ is a primitive rank 3 group of product action type. Assume that the subgraphs $Γ_P$ are complete. Then either { indent=6mm \item{(i)}there exist an $H$-transitive decomposition ${\cal T}=(K_m,{\cal Q})$ corresponding to a 2-transitive linear space such that ${\cal P}$ arises from ${\cal T}$ via construction 2.2.1; or \item{(ii)}for some 2-transitive normal subgroup $J$ of $H$ there exist a $J$-transitive decomposition ${\cal T}=(K_m,{\cal Q})$ corresponding to a 2-transitive linear space such that ${\cal P}={\cal P}({\cal T},{\cal R},φ)$ (as in construction 2.3.3) for some $φ$, where ${\cal R}$ is the partition of $AK_m$ in which each part contains only one arc. }
Reviewer: Alexandre A. Makhnev (Ekaterinburg)