Construction of some subgroups in black box groups ${\rm {PGL}}_2(q)$ and ${\rm{(P)SL}}_2(q)$

TL;DR

This paper presents algorithms for constructing specific subgroups within black box groups encrypting ${ m{PGL}}_2(q)$ and ${ m{(P)SL}}_2(q)$, including subgroups isomorphic to ${ m{Sym}}_4$ and subfield subgroups.

Contribution

It introduces new algorithms for identifying and constructing important subgroups in black box groups representing ${ m{PGL}}_2(q)$ and ${ m{(P)SL}}_2(q)$, enhancing computational group theory methods.

Findings

Algorithms successfully construct ${ m{Sym}}_4$ subgroups.

Algorithms identify subfield subgroups.

Applicable to groups with odd q.

Abstract

For the black box groups $X$ encrypting ${\rm{PGL}}_2(q)$, $q$ odd, we propose an algorithm constructing a subgroup encrypting ${\rm{Sym}}_4$ and subfield subgroups of $X$. We also present the analogous algorithms for black box groups encrypting ${\rm{(P)SL}}_2(q)$.