Finite groups with NR-subgroups or their generalizations

Abstract

Abstract. Yakov Berkovich investigated the following concept: a subgroup H of a finite group G is called anIn 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.

On constructing CAP-subgroups in direct products

“…First of all, a subgroup U of a group G is said to be normal-sensitive in G if for each N U , there exists an H G such that H ∩ U = N . (A normal-sensitive subgroup is also referred to as a CEP-subgroup as in [10].) Second, a subgroup U of a group G is said to be normal-transitive in G if each normal subgroup of U is normal in G. Last, a subgroup U of a group G is said to be CAPtransitive in G if each CAP-subgroup of U is a CAP-subgroup of G. In [14], the author proves that normal-transitivity implies both CAP-transitivity and normal-sensitivity.…”

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

“…The notion of NR-subgroup is equivalent to the notion of normal sensitive subgroup (see [6]) which is in fact equivalent to the notion of CEP -subgroup (congruence extension property), which is even older and given in other algebraic structures (see [11] [14]). NR-subgroups and NE -subgroups were used for characterizations of soluble T -groups (see [6,15,16] Lemma 2.2). We show that some subgroup embedding properties, which have been used to characterize soluble T -groups, can also be used to characterize soluble PT -and PSTgroups.…”