On s-semipermutable subgroups of finite groups and p-nilpotency

Abstract

A subgroup H of a group is said to be s-semipermutable in G if it is permutable with every Sylow p-subgroup of G with (p, |H |) = 1. Using the concept of s-semipermutable subgroups, some new characterizations of p-nilpotent groups are obtained and several results are generalized.

Predicting the Oxygen-Binding Properties of Platinum Nanoparticle Ensembles by Combining High-Precision Electron Microscopy and Density Functional Theory

“…59 60Atoms with lower coordination numbers exhibit lower electron-density values on their surfaces (as sampled on the electrostatic isosurface) due to the well-known Smoluchowski effect 42 . This effect describes the observed[43][44][45] spreading out of electronic charge density at sharp, surface corrugations and defects resulting in loss of electron-density above low coordinated sites (see supplementary informationFigure S6). This is particularly visible when contrasting the relatively high electronic charge density on facets of the cuboctahedral nanoparticles with the lower density values on the edges and even more electron deficient vertices.…”

On s-semipermutable or s-quasinormally Embedded Subgroups of Finite Groups

“…Clearly, every s-quasinormal subgroup of G is an s-semipermutable subgroup of G, but the converse does not hold. Many authors consider minimal or maximal subgroups of a Sylow subgroup of a group when investigating the structure of G, such as in [1,2] and [5][6][7][8][9][10][11][12][13][14][15], etc. For example, in [5] Han proves the following result.…”

On $$s$$ s -semipermutable or ss-quasinormal subgroups of finite groups

“…Clearly, every s-quasinormal subgroup of G is an ssemipermutable subgroup of G, but the converse does not hold. Many authors consider minimal or maximal subgroups of a Sylow subgroup of a group when investigating the structure of G, such as in [1] and [4][5][6][7][8][9][10][11][12][13][14], etc. For example, Han in [4] proves the following result.…”

“…Many authors consider minimal or maximal subgroups of a Sylow subgroup of a group when investigating the structure of G, such as in [1] and [4][5][6][7][8][9][10][11][12][13][14], etc. For example, Han in [4] proves the following result. Theorem 1.1 Let p be a prime dividing the order of a group G satisfying (|G|, p − 1) = 1 and P a Sylow p-subgroup of G. Suppose there exists a nontrivial subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H | = |D| or with order 2|D| (if P is a nonabelian 2-group and |P : D| > 2) is s-semipermutable in G, then G is p-nilpotent.…”