  Let G be a totally disconnected, locally compact group. A closed subgroup of
G is locally normal if its normaliser is open in G. We begin an investigation
of the structure of the family of closed locally normal subgroups of G. Modulo
commensurability, this family forms a modular lattice, called the structure
lattice of G. We show that G admits a canonical maximal quotient H for which
the quasi-centre and the abelian locally normal subgroups are trivial. In this
situation the structure lattice of H has a canonical subset called the
centraliser lattice, forming a Boolean algebra whose elements correspond to
centralisers of locally normal subgroups. If H is second-countable and acts
faithfully on its centraliser lattice, we show that the topology of H is
determined by its algebraic structure (and thus invariant by every abstract
group automomorphism), and also that the action on the Stone space of the
centraliser lattice is universal for a class of actions on profinite spaces.
Most of the material is developed in the more general framework of Hecke pairs.
