Titov, G. N. Lattice and group complementability in periodic locally solvable groups. (English. Russian original) Zbl 0582.20021 Algebra Logic 23, 146-155 (1984); translation from Algebra Logika 23, No. 2, 208-219 (1984). A subgroup B of a group G is called a lattice complement of a subgroup A if \(G=<A,B>\) and \(A\cap B=1\). The subgroup A is said to be \(G\Phi\)- complemented if there exists a subgroup H of G such that \(G=AH\) and \(A\cap H\subseteq \Phi (G)\), where \(\Phi\) (G) denotes the Frattini subgroup of G. A group G is said to be a \(\Phi_{\omega}C\)-group if the following conditions on a subgroup A of G imply that A is \(G\Phi\)- complemented: A is a p-subgroup of G for some prime p and A contains a Sylow p-subgroup of \(\Phi\) (G); A has a lattice complement B which contains a Sylow p’-subgroup of \(\Phi\) (G). Let G be a periodic locally solvable groups which is locally normal or has finite Sylow p-subgroups for all primes p. Theorem: if G is a \(\Phi_{\omega}C\)-group then G is locally supersolvable. There are some further related results. Reviewer: R.M.Bryant MSC: 20E25 Local properties of groups 20F50 Periodic groups; locally finite groups 20F16 Solvable groups, supersolvable groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20F24 FC-groups and their generalizations Keywords:lattice complement; Frattini subgroup; G\(\Phi \) -complemented; periodic locally solvable groups; locally normal; Sylow p-subgroups PDFBibTeX XMLCite \textit{G. N. Titov}, Algebra Logic 23, 146--155 (1984; Zbl 0582.20021); translation from Algebra Logika 23, No. 2, 208--219 (1984) Full Text: DOI EuDML References: [1] S. N. Chernikov, Groups with Given Properties of a System of Subgroups [in Russian], Nauka, Moscow (1980). [2] G. N. Titov, ”Relations between the sets of complemented and lattice complemented subgroups,” Manuscript deposited at VINITI, No. 5758–83. [3] G. N. Titov, ”Relations between the sets of complemented and lattice complemented subgroups,” in: Seventeenth All-Union Algebra Conference, Part 2, Minsk (1983), p. 234. [4] M. Suzuki, Structure of a Group and the Structure of Its Lattice of Subgroups, Springer-Verlag, Berlin (1956). · Zbl 0070.25406 [5] M. Hall, Jr., The Theory of Groups, Macmillan, New York (1959). [6] M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of the Theory of Groups [in Russian], 3rd ed., Nauka, Moscow (1982). · Zbl 0508.20001 [7] L. A. Shemetkov, Formations of Finite Groups [in Russian], Nauka, Moscow (1978). · Zbl 0496.20014 [8] A. G. Kurosh, Group Theory, Chelsea Publ. 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.