# Finite extensions of $\mathcal{H}-$ and $\mathcal{AH}-$accessible groups

## TL;DR

This paper proves that the properties of being $\\mathcal{H}-$accessible and $\\mathcal{AH}-$accessible are maintained when groups are extended by finite groups, expanding understanding of group extension stability.

## Contribution

It establishes that finite extensions preserve the properties of being $\\mathcal{H}-$accessible and $\\mathcal{AH}-$accessible in groups, a novel result in group theory.

## Key findings

- Finite extensions preserve $\\mathcal{H}-$accessibility.
- Finite extensions preserve $\\mathcal{AH}-$accessibility.
- The result applies broadly to group extension theory.

## Abstract

We prove that the group property of being $\mathcal{H}-$ and $\mathcal{AH}-$accessible is preserved under finite extensions.