$\sigma$-properties of finite groups in polynomial time

TL;DR

This paper presents polynomial-time algorithms to analyze various $\sigma$-properties of finite groups and their subgroups, including $\sigma$-nilpotency, $\sigma$-solubility, and $\sigma$-permutability, based on prime divisor partitions.

Contribution

It introduces efficient methods to determine $\sigma$-properties and find minimal partitions for finite groups and subgroups, advancing computational group theory.

Findings

Polynomial-time algorithms for $\sigma$-nilpotency and $\sigma$-solubility checks.

Methods to identify minimal partitions for $\sigma$-nilpotency and $\sigma$-permutability.

Efficient analysis of subgroup properties in permutation groups.

Abstract

Let $H, K$ be subgroups of the permutation group $G$ of degree $n$ with $K\trianglelefteq G$ and $\sigma$ be a partition of the set of all different prime divisors of $|G/K|$. We prove that in polynomial time (in $n$) one can check $G/K$ for $\sigma$-nilpotency and $\sigma$-solubility; $H/K$ for $\sigma$-subnormality and $\sigma$-$p$-permutability in $G/K$. Moreover one can find the least partition $\sigma$ of $\pi(G/K)$ for which $G/K$ is $\sigma$-nilpotent. Also one can find the least partition $\sigma$ of $\pi(G/K)$ for which $H/K$ is $\sigma$-$p$-permutable in $G/K$.