Sharply transitive sets in $\mathrm{PGL}_2(K)$

TL;DR

This paper provides a simplified proof demonstrating that every sharply transitive subset of PGL_2(K) is essentially a coset of a subgroup, clarifying the structure of these sets in projective linear groups.

Contribution

The paper offers a simplified proof of a structural property of sharply transitive sets in PGL_2(K), showing they are cosets of subgroups, which enhances understanding of their algebraic structure.

Findings

Sharply transitive subsets in PGL_2(K) are cosets of subgroups.

The proof simplifies previous arguments and clarifies the structure.

The result applies to all fields K, broadening its relevance.

Abstract

Here is a simplified proof that every sharply transitive subset of $\mathrm{PGL}_2(K)$ is a coset of a subgroup.