Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Groups with many normal or self-normalizing subgroups

In this paper groups in which the set Σ of the normal or self-normalizing subgroups is large will be studied. In particular it will be characterized locally graded groups satisfying the minimal condition on subgroups which do not belong to Σ and locally finite groups for which the set Σ is dense in the lattice of all subgroups.     

Groups whose proper subgroups are Baer groups

In this paper we classify infinite soluble minimal non-nilpotent-groups, detemine the basic structure of infinite soluble minimal non-Baer-groups, and using famed Heineken-Mohamed-groups we construct an example of minimal non-Baer-group which is not minimal non-nilpotent-group.The author would like to thank Chen Zhangmu and Shi Wuje for narm help and useful advice.     

Groups whose primary subgroups are normal sensitive

A subgroup \(H\) of a group \(G\) is said to be normal sensitive in \(G\) if for every normal subgroup \(N\) of \(H, N=H\cap N^{G}\) . In this paper we study locally finite groups whose \(p\) -subgroups are normal sensitive. We show the connection between these groups and groups in which Sylow permutability is transitive.     

Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)

Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.     

Groups whose proper factors are all abelian

The question implied in the title is a problem of A. Zaks, namely, which finite groups (other than abelian and simple groups) have all their proper factors abelian? This paper answers the question in the case of groups with non-trivial centre, or, equivalently, in the case ofp-groups, and gives a structure theorem for such groups.     

Finite p-groups all of whose non-abelian proper subgroups are metacyclic

In this paper, we classify the finite p-groups all of whose non-abelian proper subgroups are metacyclic and answer a question posed by Berkovich. Received: 22 June 2005     

Finite p-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if CG(A) = A and |NG(A)/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

Groups all of whose infinite abelian pd-subgroups are normal

The author studies groups in which any infinite Abelian pd-subgroup (p is a prime) is normal, on the assumption that the group indeed contains such subgroups (IH_p-groups). Necessary and sufficient conditions are established for a group to be an IH_p-group. Relationships are established between the class of IH_p-groups and the class of groups in which all infinite Abelian subgroups are normal, and the class of groups in which all pd-subgroups are normal.Translated from Ukrainskii Matematicheskii Zhurnal, Vo. 44, No. 6, pp. 796–800, June, 1992.