Lattice definability of matrix groups and certain other groups

Abstract

Let Z~G~ denote the lattice of the subgroups of the group~.The groups ~ and Gr are said to be lattice isomorphic if there exists an isomorphism ~ of the lattice ~G~ onto ~C~I~ If /~ is a subgroup of ~, then H f denotes the image of H under the lattice isomorphism ~ ; in particular, ~= ~IWe say that a group ~ is determined by the lattice of its subgroups or, briefly, it is lattice-determined, if ~ is isomorphic to ~I whenever ~(~ is isomorphic to ~I~r). The question of the definability of a group by the lattice of its subgroups is one of the fundamental problems in the theory of lattice isomorphisms of groups. Information on results obtained in this direction can be found in [I-4]. From more recent investigations we mention the results obtained by R. Schmidt; in his investigations [5,6] it is proved that the following groups are lattice-determined: firstly, the triply transitive and certain doubly transitive permutation groups of degree ~ , generated by involutions; secondly, the groups ~(~ over a skew field ~, where ~ ~ (for ~=~ I~I~3 ), and the groups ~(~ over a field ~ for I~I~ ; thirdly, the projective images of symplectic, orthogonal, unitary groups over a field with several additional restrictions; and, finally, the finite simple Suzuki groups, Ree groups, the Higman-Sims group, the Conway group, as well as some groups from other classes.In this paper we prove the lattice definability of groups of certain classes. If we oversimplify it somewhat, then these are the groups possessing a system V of generators such that each lattice isomorphism of the group ~1~f "~3 ~ is induced by a group isomorphism for any mutually distinct elements ~7,~z,~jE ~ , and also some groups of matrices of order ~# over an associative ring with identity, generated by transvections and, moreover, such that the additive group of the ring is either nonperiodic or is generated by elements of prime order; finally, these are some groups generated by involutions.More refined results are formulated in the theorems of Secs. 6 and 8. The proof of the theorem is carried out by a method in which the starting point consists of Theorem i.i and its corollary (Sec. i) and a fundamental role is played by Theorem 2.1 (Sec. 2). For applications Theorem 3.1 (Sec.3) is fundamental. In Secs. 4 and 5 we prove local and approximation theorems, which also play an important role in applications. -Basically, we make use of the standard notations. The meaning of the nonstandard notations is explained in the text at the appropriate place.