MODULAR SUBGROUP STRUCTURE IN INFINITE GROUPS By STEWART E. STONEHEWER [Received 2 April 1974] 1. Introduction Two groups are said to be lattice isomorphic if their subgroup lattices are isomorphic. Then it is easy to see that the image of a normal subgroup under a lattice isomorphism need not be normal. For example, the non-abelian group of order 6 and the elementary abelian group of order 9 have isomorphic subgroup lattices. However, a normal subgroup M satisfies the Dedekind modular identities Un(V,M) = (V,UnM} (1) for all subgroups U Js F, and £/n<F,Jlf> = <£/nF,Jf> (2) for all subgroups U, V with U ^ M. Moreover relations of the form (1) and (2) are clearly preserved by lattice isomorphisms. Since a subgroup M of a group G is modular in G if and only if relations (1) and (2) hold, modularity may be viewed as a natural lattice-invariant generalization of normality. Equivalently M is modular in 0 if and only if for each subgroup U of G the map [<U, M}/M] -> [U/U n M] (3) defined by F h> UnV, is a lattice isomorphism. Here if X ^ Y are subgroups of a group, [X/F] denotes the lattice of subgroups of X containing Y. Our concern with modular subgroups is to discover the extent to which they can differ from being normal. In finite groups Schmidt [8, 9] has described the structure of modular subgroups completely, modulo the structure of permutable subgroups. A subgroup H of a group G is permutable (or quasinormal) in G if HTJ = UH for all subgroups U of G. Thus H <: G implies that H is permutable in G, which in turn implies that H is modular in G. (The reverse implications do not hold in general.) The main result in [9] is the following theorem. THEOREM 1.1 (Schmidt). Let M be a core-free modular subgroup of a finite group G. Then G = P 1 xP 2 x...xP r xK, Proc. London Math. Soc. (3) 32 (1976) 63-100