Izabela Agata Malinowska

Finite groups with NR-subgroups or their generalizations

Abstract. Yakov Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if, whenever ,, where KG is the normal closure of K in G. In this article we characterize the class of finite solvable groups in which every subnormal subgroup is normal in terms of NR-subgroups. We also give similar characterizations of the classes of finite solvable groups in which every subnormal subgroup is permutable or s-permutable. Moreover we provide some sufficient conditions for the supersolvability and p-nilpotency of finite groups.

Finite groups in which normality, permutability or Sylow permutability is transitive

Abstract Y. Li gave a characterization of the class of finite soluble groups in which every subnormal subgroup is normal by means of NE-subgroups: a subgroup H of a group G is called an NE-subgroup of G if NG(H) ∩ HG = H. We obtain a new characterization of these groups related to the local Wielandt subgroup. We also give characterizations of the classes of finite soluble groups in which every subnormal subgroup is permutable or Sylow permutable in terms of NE-subgroups.

ON LATTICES OF RADICALS IN THE CLASS OF ALL FINITE GROUPS

Lattices of radicals were extensively studied, for example, in the class of associative rings and led to some interesting results. In this paper we investigate the lattice L of all radicals in the class of all finite groups. We will also consider some its important sublattices. In particular, we prove that the lattice L is closed to being modular, the lattice L_h of all hereditary radicals is a Boolean algebra, and there exists a natural, useful projection of the lattice L onto L_h. 10.1017/S0004972712000056