Elation and translation semipartial geometries

We introduce a theory of elation and translation semipartial geometries (SPG). Starting from an SPG-family ( G , J ) , i.e. a not necessarily abelian group G and a collection of subgroups J = { S 0 , … , S t } satisfying some extra condition, we construct a semipartial geometry S as a coset geometry. We show that there are strong relations between the theory of these geometries and that of elation and translation generalized quadrangles. We show for example that the theory of translation semipartial geometries is in fact almost equivalent to the study of SPG-reguli in PG ( n , q ) . We introduce a special class of automorphisms, called parallelisms, for these geometries and examine the structure of fixed points and lines under these automorphisms. In the case that G is abelian we show that in almost all cases Aut ( S ) ⩽ A Γ L ( n + 2 , q ) for certain n and q.