Infinite locally finite simple groups with many complemented subgroups

Document Type : Research Paper


1 Dipartimento di Matematica e Fisica, Universit degli Studi della Campania “Luigi Vanvitelli”, viale Lincoln 5, Caserta, Italy

2 Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universit degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, Napoli, Italy


We prove that the following families of (infinite) groups have complemented subgroup lattice‎: ‎alternating groups‎, ‎finitary symmetric groups‎, ‎Suzuki groups over an infinite locally finite field of characteristic $2$‎, ‎Ree groups over an infinite locally finite field of characteristic~$3$‎. ‎We also show that if the Sylow primary subgroups of a locally finite simple group $G$ have complemented subgroup lattice‎, ‎then this is also the case for $G$‎.


Main Subjects

[1] R. C. Alperin, PSL2 (Z) = Z2 ∗ Z3 , Amer. Math. Monthly, 100 (1993) 385–386.
[2] R. Baer, Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich,
Stud. Math., 5 (1934) 15–17.
[3] R. W. Carter, Simple groups of Lie type, Wiley, New York, 1972.
[4] M. Costantini and G. Zacher, The finite simple groups have complemented subgroup lattices, Pacific J. Math., 213
(2004) 245–251.
[5] M. Ferrara and M. Trombetti, A local study of group classes, Note Mat., 42 (2020) 1–20.
[6] M. Ferrara and M. Trombetti, Infinite groups with many complemented subgroups, Beitr. Alg. Geom, to appear;
[7] J. Hall, Infinite alternating groups as finitary linear transformation groups, J. Algebra, 119 (1988) 337–359.
[8] B. Hartley and G. Shute, Monomorphisms and direct limits of finite groups of Lie type, Quart. J. Math. Oxford (2),
35 (1984) 49–71.
[9] O. H. Kegel and B. A. F. Wehrfritz, Locally finite groups, North-Holland, London, 1973.
[10] V. M. Levchuk and Ya. N. Nuzhin, Structure of Ree groups, Algebra and Logic, 24 (1985) 16–26.
[11] A. Yu. Ol’shanskij, Geometry of defining relations in groups, Kluwer Academy Publishers, Dordrecht, 1991.
[12] E. Previato, Some families of simple groups whose lattices are complemented, Boll. U. M. I., 6 (1982) 1003–1014.
[13] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Springer, Berlin, 1972.
[14] D. J. S. Robinson, A course in the theory of groups, Springer, Berlin, 1995.
[15] R. Schmidt, Subgroup lattices of groups, De Gruyter, Berlin, 1994.
[16] M. Suzuki, On a class of doubly transitive groups, Ann. Math., 75 (1962) 105–145.
[17] H. N. Ward, On Ree’s series of simple groups, Trans. Amer. Math. Soc., 121 (1966) 62–89.
[18] B. A. F. Wehrfritz, Infinite linear groups, Springer, Berlin, 1973.