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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
##### MSC:
 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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