GAG 2019

Nilpotent groups and Euler characteristic

It is well-known that a finite group G is nilpotent if and only if G has exactly one p-Sylow subgroup for each prime p. We show the correspondent result for a family of groups that are first-order definable in some particular structures where p-groups are meant with respect to a geometric invariant, called o-minimal Euler characteristic, that plays the same role than cardinality for finite groups. As a consequence, one can find structural decompositions of such nilpotent groups, and show that each of them is elementarily equivalent to a linear algebraic group.