On the replacement property for PSL(2,p)

Abstract

The replacement property (or Steinitz Exchange Lemma) for vector spaces has a natural analog for finite groups and their generating sets. For the special case of the groups $\text{PSL}(2,p),$ where p is a prime larger than 5, first partial results concerning the replacement property were published by Benjamin Nachman (2014). Second partial results were published by Hy P.G. Lam (2017). The main goal of this paper is to provide a complete answer for $\text{PSL}(2,p).$

Keywords:

• Exchange lemma
• finite groups
• minimal generating sets
• minimax sets
• PSL(2, p)
• replacement property

2020 Mathematics Subject Classification:

• 20F05
• 20G40

Acknowledgement

The authors are thankful to R. K. Dennis for patiently and carefully teaching us the requisite information needed for this paper and for guiding our inquiries in fruitful directions. Further thanks goes to the math department at Cornell University for hosting the SPUR/REU program and of course to our group mates without whom progress would have been far slower and less enjoyable. Finally, we would like to thank the referee for providing useful comments on how to improve the paper.

Additional information

Funding

The first author is grateful to María Cristina Masaveu Peterson Foundation for their funding. The second author would also like to acknowledge and thank the Science Scholars Program at Temple University for summer funding. The third author would like to thank The Crossing Paths for their traveling grant.