The Significance of the System of Subgroups for the Structure of the Group

Abstract

Theorems in group theory whose proofs are effected by manipulating subgroups and not elements-such theorems generally refer to direct decompositions-are as a rule easily recognized as special cases of theorems in the theory of lattices. There are other theorems in the theory of groups where such a generalization is not quite obvious. Thus the problem arises to characterize those parts of the theory of groups in which the results may be expected to be special cases of theorems in the theory of lattices. Such a part of the theory of groups contains but one essentially group-theoretical statement, namely the assertion that all the facts in this theory are special cases of results in the theory of lattices. If the group G is isomorphic to every group with an isomorphic lattice of subgroups, then it may be said that the structure theory of this group G forms part of the theory of lattices. If in addition every of the lattice of the subgroups of G is induced by an of G, then the relative structure of the subgroups of G presents a problem which belongs to the theory of lattices. Thus it will be our problem to discuss the relations between the isomorphisms of groups on the one hand and the isomorphisms of the lattice of its subgroups on the other hand. The term isomorphism of the lattice of subgroups will be used in a more or less restricted sense. In its broadest meaning this term refers only to lattice properties in the accepted sense of the word, whereas the isomorphisms of the lattice of the subgroups in the more restricted sense of the term shall preserve properties like normality and numbers like the index. The modern development of the theory of lattices has been preceded by a discussion of the above problem. It has been proved in 1928 by A. Rottlaender that there exists -finite non-isomorphic groups whose lattices of subgroups are isomorphic in a rather strong sense.' Since these groups are -both not abelian,