Lattice homomorphisms induced by group homomorphisms

Abstract

Introduction. By a lattice homorphism of a group G onto a group G' we mean a single-valued mapping sb of the lattice L(G) of subgroups of G onto the lattice L(G') of subgroups of G', which preserves all unions and intersections, that is, which is subject to the conditions 1. (U,S,)(k U= (S,(k), 2. (nfS,)ck w= s, for every (finite or infinite) set of subgroups S, of G. We call proper any lattice homomorphism which is neither a lattice isomorphism (1-1), nor a trivial lattice homomorphism (So = 1 for every S in L(G)). The general problem which now presents itself is the characterization of those (subgroup) lattice mappings which are proper lattice homomorphisms, and of those groups which admit proper lattice homomorphisms (cf. Whitman [1],2 Zappa [1] and [2], and Suzuki [1]). A specialization of this problem arises when we consider a lattice mapping which is induced by a (group) homomorphism. It is easily seen that such a mapping preserves unions. Under what conditions will it be a (proper) lattice homomorphism? The main concern of this note is the characterization of those homomorphisms which induce proper lattice homomorphisms, and of those groups which admit such homomorphisms. A further specialization occurs in connection with a method suggested by G. Zappa for constructing lattice mappings which preserve intersections. Let H be a proper subgroup of a group G. Then the mapping ,6 of L(G) onto L(H) defined by