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Lattice determinability of matrix groups and certain other groups. (English. Russian original) Zbl 0666.20014
Algebra Logic 25, No. 6, 439-469 (1986); translation from Algebra Logika 25, No. 6, 696-744 (1986).
Let G be any group. Denote by L(G) the lattice of its subgroups. The group G is called lattice determined if the condition \(L(G)\simeq L(G_ 1)\) implies \(G\simeq G_ 1\). In the paper is proved that some classes of groups are lattice determined. Let the group G be generated by elements \(a_ 1,...,a_ n\), \(n\geq 4\) and defined in these generators by one defining relation \(w=1\). If the word w contains at least three generators, then the group G is lattice determined. Let R be an associative ring with identity element, \(EL_ n(R)\) be the subgroup of the general linear group \(GL_ n(R)\) generated by all elementary matrices. If the additive group of the ring R either is nonperiodic or it is generated by elements of prime orders, then the group G is lattice determined. A number of similar results is also proved. The proofs are based on techniques which are developed in the paper. The concept of a basic mapping of some subset of G in a group \(G_ 1\) with respect to a lattice isomorphism is fundamental. Some theorems (e.g. th. 1.1, th. 1.2) are proved permitting in some cases to establish the existence of a group isomorphism.
Reviewer: V.Ya.Bloshchitsyn
20E15 Chains and lattices of subgroups, subnormal subgroups
20F05 Generators, relations, and presentations of groups
20G35 Linear algebraic groups over adèles and other rings and schemes
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