Generating Sequences of PSL(2,p)

TL;DR

This paper extends the understanding of the size of irredundant generating sets in PSL(2,p), showing most have size three for a broad class of primes and analyzing element orders within these sets.

Contribution

It generalizes previous results on the maximal size of irredundant generating sets in PSL(2,p) to a larger prime class and characterizes possible element orders.

Findings

For primes p not congruent to ±1 mod 10, m(G)=3 except p=7.

Determined possible element orders in irredundant generating sets of size ≤4.

Discussed the replacement property in PSL(2,p).

Abstract

Julius Whiston and Jan Saxl showed that the size of an irredundant generating set of the group G=PSL(2,p) is at most four and computed the size m(G) of a maximal set for many primes. We will extend this result to a larger class of primes, with a surprising result that when p\not\equiv\pm 1\mod 10, m(G)=3 except for the special case p=7. In addition, we will determine which orders of elements in irredundant generating sets of PSL(2,p) with lengths less than or equal to four are possible in most cases. We also give some remarks about the behavior of PSL(2,p) with respect to the replacement property for groups.