Some Structural Properties of Upper and Lower Central Series of Pairs of Groups (original) (raw)

Characterizations of nested GVZ-groups by central series

arXiv: Group Theory, 2019

Many properties of groups can be defined by the existence of a particular normal series. The classic examples being solvability, supersolvability and nilpotence. Among the nilpotent groups are the so-called nested GVZ-groups --- groups where the centers of the irreducible characters form a chain, and where every irreducible character vanishes off of its center. In this paper, we show that nested GVZ-groups can be characterized by the existence of a certain ascending central series, or by the existence of a certain descending central series.

Isoclinic Classification of Some Pairs (G,G′)(G,G')(G,G) of ppp-Groups

Iranian Journal of Mathematical Sciences and Informatics, 2018

The equivalence relation isoclinism partitions the class of all pairs of groups into families. In this paper, a complete classification of the set of all pairs (G,G) is established whenever, p is a prime number greater than 3 and G is a p-group of order at most p. Moreover, the classification of pairs (H,H) for extra special p-groups H is also given.

Group extensions and marginal series of pair of groups

Comptes Rendus. Mathématique, 2021

In this article, using the concept of generalized Baer-invariant of a pair of groups, we establish some related isomorphisms between lower marginal quotient pairs of groups, which are generalized versions of some isomorphisms of Stallings. We also derive a result for the pair (V .W , X) to be an ultra Hall pair for special varieties of groups. This result generalizes that of Fung in 1977, which has roots in Philip Hall's criterion on nilpotency.

A characterization of Nested Groups in terms of conjugacy classes

Comptes Rendus. Mathématique, 2020

A group is nested if the centers of the irreducible characters form a chain. In this paper, we will show that there is a set of subgroups associated with the conjugacy classes of group so that a group is nested if and only if these subgroups form a chain.

A note on finite 𝒫𝒮𝒯-groups

Journal of Group Theory, 2007

A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.

On the Nilpotency of a Pair of Groups

Southeast Asian Bulletin of …, 2012

This paper is devoted to suggest that the extensive theory of nilpotency, upper and lower central series of groups could be extended in an interesting and useful way to a theory for pairs of groups. Also this yields some information on nilpotent groups.

Some Properties on Isologism of Groups

Journal of the Australian Mathematical Society, 2000

In this paper a necessary and sufficient condition will be given for groups to be V-isologic, with respect to a given variety of groups V. It is also shown that every V-isologism family of a group contains a V-Hopfian group. Finally we show that if G is in the variety V, then every V-covering group of G is a Hopfian group.