The Replacement Property of PSL$(2,p)$ and PSL$(2,p^2)$

TL;DR

This paper extends the classification of the replacement property in PSL(2,q) groups for q=p and p^2, considering conditions on p modulo 8 and 10, building on previous work by Nachman.

Contribution

It generalizes the classification of the replacement property for PSL(2,q) groups with q=p and p^2 under new modular conditions on p.

Findings

Classification of the replacement property for PSL(2,p) and PSL(2,p^2) under specified modular conditions.

Identification of conditions on p that determine whether the group satisfies or fails the replacement property.

Extension of Nachman's results to broader cases involving prime powers and modular constraints.

Abstract

In 2014, Benjamin Nachman showed that when $p\equiv$1 mod 8, the 2-dimensional projective linear group over the field of $p$ elements fails the replacement property if the maximal length $m$ of an irredundant generating sequence for the group is 3. In addition, if $m=4$, the group satisfies the property for any prime $p$. In this paper, we will extend such classification for PSL$(2,q)$ where $q$ is $p$ and $p^2$ with conditions of modulo 8 and 10 on $p$.