On the Lattice of Subgroups of Finite Groups

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 P-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 P-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]('), 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 P-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 P-isomorphic to a ¿>-group(2). In §4 we consider the image 4>(S) of a £-Sylow subgroup S of a group G by an P-isomorphism from G onto a group H, and treat the case when -Sylow subgroup of H. In §5 we deal with the similar problem for the image of normal subgroups by an P-isomorphism. After these considerations we prove that groups P-isomorphic to a solvable group are solvable, too, and that a group G P-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 P-isomorphisms of finite groups. Some of our results may be generalized, however, to the case of P-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