The supersolvability of QCLT groups
Copyright policy )The supersolvability of QCLT groups
Only finite groups are considered. A group is said to be CLT if every divisor of its order is the order of some subgroup. A group all of whose homomorphic images are CLT is said to be QCLT. Any supersolvable group is QCLT but the converse is false. \textit{J. F. Humphreys} [Proc. Camb. Philos. Soc. 75, 25-32 (1974; Zbl 0273.20018)] gave a partial converse that any QCLT group of odd order is supersolvable. In an earlier work (appeared later!) [Acta Math. Sin., New Ser. 2, 78-81 (1986)] the present author has observed that under certain general conditions a group G has a unique minimal normal subgroup F(G) and a maximal subgroup A which is a complement of F(G) in G having some properties. Using that observation he now gives several sets of conditions under each of which a QCLT group is supersolvable; one such is the Corollary 5 that a QCLT group whose commutator subgroup is of odd order is supersolvable.
- Peking University China (People's Republic of)
- Peking University China (People's Republic of)
QCLT group, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, Series and lattices of subgroups, maximal subgroup, Special subgroups (Frattini, Fitting, etc.), commutator subgroup, Finite nilpotent groups, \(p\)-groups, complement, minimal normal subgroup, supersolvable group, Arithmetic and combinatorial problems involving abstract finite groups, Subnormal subgroups of abstract finite groups
QCLT group, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, Series and lattices of subgroups, maximal subgroup, Special subgroups (Frattini, Fitting, etc.), commutator subgroup, Finite nilpotent groups, \(p\)-groups, complement, minimal normal subgroup, supersolvable group, Arithmetic and combinatorial problems involving abstract finite groups, Subnormal subgroups of abstract finite groups
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