five_transitive_card_classification.toml

id = "five_transitive_card_classification"
title = "Possible orders of 5-transitive finite permutation groups"
test = false
module = "LeanEval.GroupTheory.MultiplyTransitive"
holes = ["five_transitive_card_classification"]
submitter = "Kim Morrison"
notes = "If a finite group acts faithfully and 5-transitively on a set X with |X| ≥ 5, then |G| is one of n!, n!/2 (only when n ≥ 7), 95040 (= |M₁₂|), or 244823040 (= |M₂₄|). The 5 ≤ |X| hypothesis prevents the 5-transitivity condition from being vacuously satisfied (otherwise small groups like C₃ on Fin 3 would qualify). Classification is folklore via CFSG; no CFSG-free proof is known."
source = "Folklore via CFSG; classical work of Mathieu, Jordan; modern accounts in P. Cameron, Permutation Groups (1999)."
informal_solution = "By CFSG, the only finite 2-transitive groups are explicitly classified. Restricting to 5-transitive: the symmetric group Sₙ is k-transitive for all k ≤ n; the alternating group Aₙ is k-transitive for k ≤ n − 2 (so 5-transitive for n ≥ 7); among the Mathieu groups, M₁₂ is sharply 5-transitive on 12 points and M₂₄ is 5-transitive on 24 points. M₁₁ and M₂₃ are only 4-transitive and so do not appear. No other finite simple group has a 5-transitive permutation representation."