Transitive permutation groups where nontrivial elements have at most two fixed points

Transitive permutation groups with trivial four point stabilizers

“…|Gω| } and hence |F Ω (x)| ≤ 3. Now |G ω | ∈ {q − 1, q + 1} and so (G, Ω) appears in the conclusion of the main theorem of [19]. In particular no nonidentity element of G has three fixed points on Ω, contrary to our assumption.…”

“…As in the proof of Lemma 4.2 this implies that G ω is cyclic of order (q ± 1)/2. This action occurs in the conclusion of the main theorem of [19], contradicting Hypothesis 2.4. Now if r ∈ π(G ω ) and (q, r) = 1, then r ⊢ p for all divisors p of (q − 1)/2.…”

“…Then G γ ∩ E(G) contains elements of order 3 and 2, which is impossible. [19] applies. We refer to Theorem 5.6 in the same paper for details on the possible action of E on ∆.…”

“…Now any subfield group contains Sz (2), a group of order 20, and then Lemma 2.2 implies that E α has an element of order 5. But we know from [19] that |E α | 2 ′ = (q − 1). Since (q − 1) is not divisible by 5 (because q is a power of 2 with odd exponent), we see that E cannot be a Suzuki group.…”

“…The latter possibility is ruled out because no involution in G ω inverts x, and the possibility G ω = B is ruled out because the action of G on the set of cosets of B occurs in the conclusion of the main theorem of [19]. Thus G ω is cyclic of order (q − 1)/2 but again this possibility occurs in the conclusion of the main theorem of [19]. This proves that PSL 2 (q) for q ≥ 13 does not yield examples satisfying Hypothesis 2.4.…”