The geometry of diagonal groups

Diagonal groups are one of the classes finite primitive permutation occurring in conclusion O’Nan–Scott theorem. Several other have been described as automorphism geometric or combinatorial structures such affine spaces Cartesian decompositions, but for diagonal not studied general. The main purpose this paper is to describe and characterise structures, which we call semilattices. Unlike theorem, defined over characteristically simple groups, our construction works arbitrary infinite. A semilattice depends on a dimension $m$ group $T$. For $m=2$, it Latin square, Cayley table $T$, though fact any square satisfies axioms. However, $m \geqslant3$, $T$ emerges naturally uniquely from (The situation somewhat resembles projective geometry, where planes exist great profusion higher-dimensional coordinatised by an algebraic object, division ring.) contained partition lattice set $\Omega$, provide introduction calculus partitions. Many concepts constructions come experimental design statistics. We also determine when can be primitive, quasiprimitive (these conditions turn out equivalent groups). Associated with graph, has same except four small cases $m\leqslant 3$. class graphs includes some well-known families, Latin-square folded cubes, potentially interest. obtain partial results chromatic number mention application synchronization property groups.