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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
Full Text: DOI
[1] 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 · doi:10.2307/2371383
[2] M. Suzuki, Structure of a Group and the Structure of Its Lattice of Subgroups, Springer, Berlin (1956). · Zbl 0070.25406
[3] L. E. Sadovskii, ”Some lattice-theoretic questions of group theory,” Usp. Mat. Nauk,23, No. 3, 122–157 (1968).
[4] M. N. Arshinov and L. E. Sadovskii, ”Some lattice-theoretic properties of groups and semigroups,” Usp. Mat. Nauk,27, No. 6, 139–180 (1972).
[5] R. Schmidt, ”Untergruppenverbande zweifach transitiver Permutationsgruppen,” Math. Z.,144, 161–168 (1975). · Zbl 0302.20002 · doi:10.1007/BF01190945
[6] R. Schmidt, ”Untergruppenverbande involutorisch erzeugter Gruppen,” Rend. Sem. Mat. Univ. Padova,63, 95–126 (1980). · Zbl 0454.20037
[7] L. E. Sadovskii (L. E. Szadovsky), ”Uber die Strukturenisomorphismen von Freigruppen,” Dokl. Akad. Nauk SSSR, No. 3,32, 171–174 (1941). · Zbl 0061.02501
[8] L. E. Sadovskii, ”Projectivities and isomorphisms of nilpotent groups,” Izv. Akad. Nauk SSSR, Ser. Mat.,29, No. 1, 171–208 (1965).
[9] A. S. Pekelis, ”Lattice isomorphisms of mixed metabelian groups,” Sib. Mat. Zh.,8, No. 4, 827–834 (1967).
[10] A. L. Shmel’kin, ”Free polynilpotent groups,” Dokl. Akad. Nauk SSSR,151, No. 1, 73–75 (1963).
[11] L. E. Sadovskii, ”An approximation theorem and lattice isomorphisms”, Dokl. Akad. Nauk SSSR,161, No. 2, 300–303 (1965).
[12] 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).
[13] W. Magnus, ”Uber diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz),” J. Reine Angew. Math.,163, 141–165 (1931). · JFM 56.0134.03
[14] K. Murasugi, ”The center of a group with a single defining relation,” Math. Ann.,155, 246–251 (1964). · Zbl 0119.02601 · doi:10.1007/BF01344162
[15] L. E. Sadovskii, ”On lattice isomorphisms of free products of groups,” Mat. Sb.,21 (63), No. 1, 63–82 (1947). · Zbl 0038.01201
[16] C. S. Holmes, ”Projectivities of free products,” Rend. Sem. Mat. Univ. Padova,42, 341–387 (1969). · Zbl 0245.20031
[17] M. N. Arshinov, ”Projectivities of a free product and isomorphisms of groups,” Sib. Mat. Zh.,11, No. 1, 12–19 (1970). · Zbl 0203.32302
[18] D. Gorenstein, Finite Simple Groups. An Introduction to Their Classification, Plenum Press, New York (1982). · Zbl 0483.20008
[19] 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
[20] B. V. Yakovlev, ”Lattice isomorphisms of metabelian groups of exponent p,” Mat. Zametki Krasnoyarsk. Pedagog Inst.,1, 59–68 (1965).
[21] A. S. Pekelis, ”Lattice isomorphisms of mixed nilpotent groups,” Sib. Mat. Zh.,6, 1315–1321 (1965).
[22] B. V. Yakovlev, ”Lattice isomorphisms of alternating groups,” in: 14th All-Union Algebra Conference, Novosibirsk (1977).
[23] B. V. Yakovlev, ”Lattice isomorphisms of certain groups, generated by involutions,” in: 6th All-Union Symposium on Group Theory, Kiev (1978).
[24] B. V. Yakovlev, ”Lattice isomorphisms of groups with a single defining relation,” in: 8th All-Union Symposium on Group Theory, Kiev (1982).
[25] B. V. Yakovlev, ”Lattice definability of groups of a certain class,” in: 9th All-Union Symposium on Group Theory, Moscow (1984).
[26] 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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