On $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups of finite groups and related formations

TL;DR

This paper introduces and studies the properties of $ ext{K}$-$ ext{P}_t$-subnormal subgroups in finite groups, exploring their structural implications and related group classes, extending subgroup normality concepts with prime divisibility conditions.

Contribution

It defines $ ext{K}$-$ ext{P}_t$-subnormal subgroups and investigates their properties and the classes of groups containing such subgroups, advancing the understanding of subgroup normality with prime-related conditions.

Findings

Characterization of $ ext{K}$-$ ext{P}_t$-subnormal subgroups

Properties of groups with Sylow $ ext{K}$-$ ext{P}_t$-subnormal subgroups

Structural implications for finite groups

Abstract

Let $t$ be a fixed natural number. A subgroup $H$ of a group $G$ will be called $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal in $G$ if there exists a chain of subgroups $H = H_{0} \leq H_{1} \leq \cdots \leq H_{m-1} \leq H_{m} = G$ such that either $H_{i-1}$ is normal in $H_{i}$ or $|H_{i} : H_{i-1}|$ is a some prime $p$ and $p-1$ is not divisible by the $(t+1)$th powers of primes for every $i = 1,\ldots , n$. In this work, properties of $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups and classes of groups with Sylow $\mathrm{K}$-$\mathbb{P}_{t}$-subnormal subgroups are obtained.