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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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##### References:
  R. Baer, ”The significance of the system of subgroups for the structure of the group,” Am. J. Math.,61, 1–44 (1939). · JFM 65.0060.01  M. Suzuki, Structure of a Group and the Structure of Its Lattice of Subgroups, Springer, Berlin (1956). · Zbl 0070.25406  L. E. Sadovskii, ”Some lattice-theoretic questions of group theory,” Usp. Mat. Nauk,23, No. 3, 122–157 (1968).  M. N. Arshinov and L. E. Sadovskii, ”Some lattice-theoretic properties of groups and semigroups,” Usp. Mat. Nauk,27, No. 6, 139–180 (1972).  R. Schmidt, ”Untergruppenverbande zweifach transitiver Permutationsgruppen,” Math. Z.,144, 161–168 (1975). · Zbl 0302.20002  R. Schmidt, ”Untergruppenverbande involutorisch erzeugter Gruppen,” Rend. Sem. Mat. Univ. Padova,63, 95–126 (1980). · Zbl 0454.20037  L. E. Sadovskii (L. E. Szadovsky), ”Uber die Strukturenisomorphismen von Freigruppen,” Dokl. Akad. Nauk SSSR, No. 3,32, 171–174 (1941). · Zbl 0061.02501  L. E. Sadovskii, ”Projectivities and isomorphisms of nilpotent groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,29, No. 1, 171–208 (1965).  A. S. Pekelis, ”Lattice isomorphisms of mixed metabelian groups,” Sib. Mat. Zh.,8, No. 4, 827–834 (1967).  A. L. Shmel’kin, ”Free polynilpotent groups,” Dokl. Akad. Nauk SSSR,151, No. 1, 73–75 (1963).  L. E. Sadovskii, ”An approximation theorem and lattice isomorphisms”, Dokl. Akad. Nauk SSSR,161, No. 2, 300–303 (1965).  B. V. Yakovlev, ”On conditions under which a lattice is isomorphic to the lattice of subgroups of a group,” Algebra Logika,13, No. 6, 694–712 (1974).  W. Magnus, ”Uber diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz),” J. Reine Angew. Math.,163, 141–165 (1931). · JFM 56.0134.03  K. Murasugi, ”The center of a group with a single defining relation,” Math. Ann.,155, 246–251 (1964). · Zbl 0119.02601  L. E. Sadovskii, ”On lattice isomorphisms of free products of groups,” Mat. Sb.,21 (63), No. 1, 63–82 (1947). · Zbl 0038.01201  C. S. Holmes, ”Projectivities of free products,” Rend. Sem. Mat. Univ. Padova,42, 341–387 (1969). · Zbl 0245.20031  M. N. Arshinov, ”Projectivities of a free product and isomorphisms of groups,” Sib. Mat. Zh.,11, No. 1, 12–19 (1970). · Zbl 0203.32302  D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, New York (1982). · Zbl 0483.20008  R. F. Spring, ”Lattice isomorphisms of finite non-abelian groups of exponent p,” Proc. Am. Math. Soc.,14, No. 3, 407–413 (1963). · Zbl 0115.25304  B. V. Yakovlev, ”Lattice isomorphisms of metabelian groups of exponent p,” Mat. Zametki Krasnoyarsk. Pedagog Inst.,1, 59–68 (1965).  A. S. Pekelis, ”Lattice isomorphisms of mixed nilpotent groups,” Sib. Mat. Zh.,6, 1315–1321 (1965).  B. V. Yakovlev, ”Lattice isomorphisms of alternating groups,” in: 14th All-Union Algebra Conference, Novosibirsk (1977).  B. V. Yakovlev, ”Lattice isomorphisms of certain groups, generated by involutions,” in: 6th All-Union Symposium on Group Theory, Kiev (1978).  B. V. Yakovlev, ”Lattice isomorphisms of groups with a single defining relation,” in: 8th All-Union Symposium on Group Theory, Kiev (1982).  B. V. Yakovlev, ”Lattice definability of groups of a certain class,” in: 9th All-Union Symposium on Group Theory, Moscow (1984).  B. V. Yakovlev, ”On lattice definability of special linear groups over a ring,” in: 18th All-Union Algebra Conference, Vol. 2, Kishinev (1985).
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