×

On subsemigroup lattices of aperiodic groups. (English) Zbl 0812.20013

The paper concerns subsemigroups of aperiodic groups. The main results are the following two theorems: Theorem 1. There are continuously many non-isomorphic 2-generated torsion-free groups each group \(G\) of which has the properties that every maximal subsemigroup of \(G\) is a cyclic subgroup of \(G\) and different maximal subgroups of \(G\) intersect trivially. Theorem 2. Given a prime \(n \gg 1\) (e.g. \(n > 10^{80}\)) there are continuously many non-isomorphic 2-generated aperiodic groups each group \(G\) of which has the following two properties: (a) the semigroup law \(x^ n(x^ n y^ n)^ n = (x^ n y^ n)^ n x^ n\) holds in \(G\); (b) every maximal subsemigroup of \(G\) is a cyclic subgroup of \(G\) either of infinite order or of order \(n\) and different maximal subgroups of \(G\) intersect trivially.

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20M05 Free semigroups, generators and relations, word problems
20E28 Maximal subgroups
20E34 General structure theorems for groups
20E07 Subgroup theorems; subgroup growth
20E10 Quasivarieties and varieties of groups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Adian, S. I., ”The Burnside problem and identities in groups”, Nauka, Moscow, 1975 (Russian). · Zbl 1285.03047
[2] Arshinov, M. N.,On lattice isomorphisms of mixed nilpotent groups, Trudy Zon. Objed. Mat. Kafedri Pedagog. Inst. Sibiri2 (1972), 8–29 (Russian).
[3] Ivanov, S. V.,On group rings of Noetherian groups, Mathematical Notes46 (1989), 929–933. · Zbl 0702.16017
[4] Petropavlovskaya, R. V.,Associative systems which are lattice-isomorphic to groups I, II, III, Vestnik Leningrad. Univ.13 (1956), 5–25,19 (1956), 80–99,19 (1957), 5–19 (Russian). · Zbl 0105.01601
[5] Petropavlovskaya, R. V.,On a certain class of groups which are determined by subsemigroup lattices, Math. Sbornik66 (1965), 256–271 (Russian).
[6] Ol’shanskii, A. Yu.,On groups with cyclic subgroups, Bull. Bulg. Akad. Sciences32 (1979), 1165–1166 (Russian). · Zbl 0431.20026
[7] Ol’shanskii, A. Yu.,On a geometric method in combinatorial group theory, Proc. Int. Congress Math.1 (1983), 415–424.
[8] Ol’shanskii, A. Yu., ”Geometry of defining relations in groups”, Nauka, Moscow, 1989 (Russian).
[9] Shevrin, L. N. and A. J. Ovsyannikov,Semigroups and their subsemigroup lattices, Semigroup Forum27 (1983), 1–154. · Zbl 0523.20037 · doi:10.1007/BF02572737
[10] Shevrin, L. N. and A. J. Ovsyannikov,Finitely assembled semigroups and ascending chain condition for subsemigroups, Proc. of Monash Conf. (1990), (to appear). · Zbl 0971.20505
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.