M. F. Ragland

Generalizations of Groups in which Normality Is Transitive

A group G is called a Hall𝒳-group if G possesses a nilpotent normal subgroup N such that G/N′ is an 𝒳-group. A group G is called an 𝒳o-group if G/Φ(G) is an 𝒳-group. The aim of this article is to study finite solvable Hall𝒳-groups and 𝒳o-groups for the classes of groups 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯. Here 𝒯, 𝒫𝒯, and 𝒫𝒮𝒯 denote, respectively, the classes of groups in which normality, permutability, and Sylow-permutability are transitive relations. Finite solvable 𝒯-groups, 𝒫𝒯-groups, and 𝒫𝒮𝒯-groups were globally characterized, respectively, in Gaschütz (1957), Zacher (1964), and Agrawal (1975). Here we arrive at similar characterizations for finite solvable Hall𝒳-groups and 𝒳o-groups where 𝒳 ∈ {𝒯, 𝒫𝒯, 𝒫𝒮𝒯}. A key result aiding in the characterization of these groups is their possession of a nilpotent residual which is a nilpotent Hall subgroup of odd order. The main result arrived at is Hall𝒫𝒮𝒯 = 𝒯o for finite solvable groups.

A note on finite PST-groups

Abstract 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 the classes of finite 𝒫𝒮𝒯-groups and finite 𝒯*-groups coincide. A new characterization of soluble 𝒫𝒮𝒯-groups is also presented.

ON A CLASS OF SUPERSOLUBLE GROUPS

A subgroup \(H\) of a finite group \(G\) is said to be S-semipermutable in \(G\) if \(H\) permutes with every Sylow \(q\)-subgroup of \(G\) for primes \(q\) not dividing \(|H\)|. A finite group \(G\) is an MS-group if the maximal subgroups of all the Sylow subgroups of \(G\) are S-semipermutable in \(G\). The aim of the present paper is to characterise the finite MS-groups. DOI: 10.1017/S0004972714000306

On hypercentral factor groups from certain classes

Throughout the rest of this paper, the hypercenter, nilpotent residual, Fitting subgroup and Frattini subgroup of G will be denoted by Z ðGÞ, g ðGÞ, FitðGÞ and FðGÞ respectively. In [18] the first two authors introduce a new class of groups, the socalled T1-groups. A group G is a T1-group if G=Z ðGÞ is a T-group. Some of the basic properties of T1-groups are established in [18]. The present work is a continuation of [18]. In the light of [18, Theorem A] and Theorem 1, we begin with

Some Characterisations of Soluble SST-Groups

All groups considered in this paper are finite. A subgroup H of a group G is said to be SS-permutable or SS-quasinormal in G if H has a supplement K in G such that H permutes with every Sylow subgroup of K. Following [6], we call a group G an SST-group provided that SS-permutability is a transitive relation in G, that is, if A is an SS-permutable subgroup of B and B is an SS-permutable subgroup of G, then A is an SS-permutable subgroup of G. The main aim of this paper is to present several characterisations of soluble SST-groups.

Corrigendum. A note on finite -groups

This paper has been published in Journal of Group Theory, 12(6):961-963 (2009). Copyright 2009 by Walter de Gruyter. The final publication is available at www.degruyter.com. http://dx.doi.org/10.1515/JGT.2009.026 http://www.degruyter.com/view/j/jgth.2009.12.issue-6/jgt.2009.026/jgt.2009.026.xml

Some local properties defining To-groups and related classes of groups

We call G a HallX -group if there exists a normal nilpotent subgroup N of G for which G=N 0 is an X -group. We call G a T0-group provided G=( G) is a T -group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally dene Hall X -groups and T0-groups where X 2 fT ;PT ;PST g; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations. 2010 Mathematics Subject Classication: Primary: 20D10, 20D20, 20D35.

Finite groups in which pronormality and -pronormality coincide

Abstract For a formation 𝔉, a subgroup U of a finite group G is said to be 𝔉-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉𝔉 such that Ux = Ug. If 𝔉 contains 𝔑, the formation of nilpotent groups, then every 𝔉-pronormal subgroup is pronormal and, in fact, 𝔑-pronormality is just classical pronormality. The main aim of this paper is to study classes of finite soluble groups in which pronormality and 𝔉-pronormality coincide.

MINI REVIEWSome subgroup embeddings in finite groups: A mini review☆

In this survey paper several subgroup embedding properties related to some types of permutability are introduced and studied.

Some subgroup embeddings in finite groups: A mini review☆

In this survey paper several subgroup embedding properties related to some types of permutability are introduced and studied.