A certain property of abelian groups
Let be a group. If every nontrivial subgroup of has a proper supplement, then is called an -group. We study some properties of -groups. For instance, it is shown that a nilpotent group is an -group if and only if is a subdirect product of cyclic groups of prime orders. We prove that if is an -group which satisfies the descending chain condition on subgroups, then is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group is an -group....
A θ-pair for a maximal subgroup M of a group G is a pair (A, B) of subgroups such that B is a maximal G-invariant subgroup of A with B but not A contained in M. θ-pairs are considered here in some groups having supersoluble maximal subgroups.
A simple proof is given of a well-known result of the existance of lattice-isomorphisms between locally nilpotent quaternionfree modular groups and abelian groups.
Let be any group and let be an abelian quasinormal subgroup of . If is any positive integer, either odd or divisible by , then we prove that the subgroup is also quasinormal in .