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The study of group properties and group structure can proceed from various initial principles. Usually, properties of a group are obtained from the properties of its elements. However, in a number of investigations essentially the converse problems are solved: to study the influence of the properties of the set of all subgroups of a group on the properties of its elements. This explains to what extent purely set-theoretical operations given on the set of all subgroups of a group G (this set forms a lattice S(G)) determine the properties of the group operation itself. The investigation of connections between the structure of a group and the structure of its lattice of subgroups has become one of the main general approaches in the study of groups. There are many interesting results in this field, but also many important unsolved problems. These problems seem to be difficult, and apparently new ideas and mathematical techniques are needed for their solution. The present survey (without claiming to exhaust the whole material) is devoted to the so-called lattice-theoretical questions of group theory and contains recent results. The main attention is given to infinite groups. The structures of various classes of finite groups, their lattice isomorphisms and homomorphisms, and many related questions are treated in Suzuki's comprehensive monograph [1]. General surveys of the results (up to 1961) for infinite soluble, nilpotent and radical groups are contained in the papers of Kontorovich, Pekelis, and Starostin [2] and Plotkin [3]. Finally, the basic notions (for group theory see [4], [5], for lattice theory [6]) and a brief survey of results in this field are contained in the new edition of Kurosh's monograph [5]. Under the circumstances there is naturally some overlap between this work and the above articles. This can be justified by the intention to make the report as complete as possible.

The Fourier-Stieltjes algebra of an arbitrary locally compact group G is the set of all finite, complex-linear combinations of continuous, positive definite functions on G, where addition and multiplication are defined pointwise and a Banach algebra norm (unique up to equivalence) can be specified. Thus, B(G) is a commutative, semisimple Banach algebra with unit. The main result is that B(G1) and B(G2) are isometrically isomorphic as Banach algebras if and only if G1 and G2 are topologically isomorphic as groups. The spectrum of B(G) is characterized as a ∗-semigroup of operators on Hilbert space, and its subgroup of invertible elements (being precisely those unitary elements which “preserve tensor products”) is topologically isomorphic to G. The Fourier algebra A(G) is also shown to characterize G. (A(G) can be defined as the closure in B(G) of the functions in B(G) with compact support.) The representation theory of the lattice of subgroups of G is also studied. The main techniques of investigation come from the theory of C ∗ and W∗-algebras.

Various characteristic conjugacy classes of subgroups having covering/avoidance properties with respect to chief factors have recently played a major role in the study of finite soluble groups. Apart from the subgroups which are now called Hall subgroups, P. Hall [7] also considered the system normalizers of a finite soluble group and showed that these form a characteristic conjugacy class, cover the central chief factors and avoid the rest. The system normalizers were later shown by Carter and Hawkes [1] to be the simplest example of a wealth of characteristic conjugacy classes of subgroups of finite soluble groups which arise naturally as a consequence of the theory of formations.

Let G be a group. We shall denote by L(G) the lattice formed by all subgroups of G. Two groups G and H are said to be lattice-isomorphic, or in short L-isomorphic, when their lattices L(G) and L(H) are isomorphic to each other. In this case the isomorphism from L(G) onto L(H) is called the L-isomorphism from G onto H. If G and H are isomorphic as groups, then G is clearly L-isomorphic to H. The converse of this statement is not always true. So the question arises: To what extent is a group determined by its lattice of subgroups? This paper is concerned with this question. The content of this paper is as follows. In ?1 we give some lemmas on groups or on lattices of subgroups. Some of these lemmas are known (cf. Ore [7](1), Iwasawa [5], and Jones [6]), others are given in generalized forms or with new demonstrations, and they are arranged so as to facilitate the following study. In ?2 we consider the number of types of groups L-isomorphic to a given group G, and prove that this number is finite if L(G) has no chain as its direct component. In ?3 we determine the structure of groups L-isomorphic to a p-group(2). In ?4 we consider the image +(S) of a p-Sylow subgroup S of a group G by an L-isomorphism 4 from G onto a group H, and treat the case when +(S) is not a p-Sylow subgroup of H. In ?5 we deal with the similar problem for the image of normal subgroups by an L-isomorphism. After these considerations we prove that groups L-isomorphic to a solvable group are solvable, too, and that a group G L-isomorphic to a perfect group H is also a perfect group with the same order as H and that, in this case, the modular lattice formed by all normal subgroups of G and H are isomorphic to each other. Moreover, we prove in ?6 that a simple group G is isomorphic to a group H if and only if L(GXG) is isomorphic to L(HXH). The two remaining sections 7 and 8 are devoted to the study of dualisms in the sense of Baer [2]. In these sections the structures of general nilpotent (finite solvable) groups with duals are completely determined. In this paper we deal chiefly with L-isomorphisms of finite groups. Some of our results may be generalized, however, to the case of L-homomorphisms of finite groups. This will be treated in the next paper. In preparing this paper the author is greatly indebted to Prof. S. Iyanaga, who made many useful comments. The author wishes to express his sincere

A group is said to have dense normal subgroups, if each non-empty open interval in its lattice of subgroups contains a normal subgroup. The structure of this and related classes of groups is investigated. Typical results are: an infinite group with dense ascendant subgroups is locally nilpotent: a nontorsion group with dense normal subgroups is abelian, etc.

Publisher Summary This chapter discusses the lattice of subgroups. The subsystems of any algebraic system form a lattice if the ordering relation of the lattice is taken to be the set-theoretic inclusion relation. The chapter focuses on the lattice of all subgroups of an abelian group and discusses the issues of the theory of subgroup lattices in this special case. The chapter focuses on addressing the following problems: (1) to find necessary and sufficient conditions for a lattice such that it should be the subgroup lattice of some group; (2) the same problem under some additional condition on the group; and (3) the description of all groups with the same lattice of subgroups, especially the determination of the cases when the subgroup lattice uniquely determines the group structure.

Publisher Summary This chapter discusses group lattices and homomorphism. A lattice of subgroups is a collection of chains of subgroups. At the top of the lattice is the head group to which every group in the lattice is a subgroup. At the bottom of the lattice is the tail group, which is a subgroup to all groups in the lattice. Associated with each Hamiltonian is a lattice of subgroups, the tail group of which commutes with the Hamiltonian and supplies exact quantum numbers. If in passing down a chain, each group commutes with a set of terms in the Hamiltonian of progressively lesser strength, the chain is said to be a physically significant one. The head group is the group of a zero-order Hamiltonian supplying a set of configurations and the quantum numbers for these configurations. The second group in the chain commutes with a portion of the Hamiltonian that is regarded as a perturbation on the zero-order term. If the perturbation is considerably weaker than the zero-order term, configuration interaction can be neglected and good eigen-values are obtained by diagonalizing the perturbation Hamiltonian inside each configuration. The diagonalization of a Hamiltonian within a configuration is simplified by symmetry adaptation of the basis of the configuration to the group of the Hamiltonian. The size of the secular equation to be solved is reduced and group-theoretical identification of the states of the Hamiltonian is made in advance.

Publisher Summary This chapter describes a program for the calculation of characters and representations of finite groups. The program described in the chapter is part of a system of programs for the investigation of finite groups. The program avoids numerical calculations as far as possible. Instead, properties of the given group that are available from other programs have been used to construct characters and representations by the process of induction. Only when this process does not yield all the required information does the program use numerical methods. The program X for the calculation of characters and representations of a given finite group G makes use of the data determined by a program Φ for the investigation of the lattice of subgroups and of certain other properties of G. This chapter also analyzes one-dimensional representations as well as irreducible representations of higher dimensions.

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 isomorphism of the lattice of the subgroups of G is induced by an isomorphism 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

The relation between two groups whose lattices of subgroups are isomorphic has been studied by various writers, including Rottlaender [5], Baer [1], Iwasawa [3], Zappa [6, 7], and Jones [4]. In the present paper we consider homomorphisms of lattices of subgroups. The principal result (Theorem 1) is: If G is a finite group and H a cyclic group, and there is a homomorphism of the lattice of subgroups of G onto the lattice of subgroups of H, then G contains a cyclic subgroup which is mapped onto H by the homomorphism.