Non-split Sharply 2-Transitive Groups of Odd Positive Characteristic

It has long been known that every sharply 2-transitive group of characteristic $3$ splits. Answering a question of Caprace, we construct the first examples of non-split sharply 2-transitive groups in odd positive characteristic p, for sufficiently large primes p. Furthermore, we show that any group without 2-torsion can be embedded into a non-split sharply 2-transitive group of characteristic p for all sufficiently large primes p, yielding 2^ ₀ many pairwise non-isomorphic countable non-split sharply 2-transitive groups in any sufficiently large characteristic. Incidentally, we answer an open question posed by Borovik and Nesin in [7].