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A361382 The orders, with repetition, of subset-transitive permutation groups.

%I A361382 #13 Mar 13 2023 05:59:15
%S A361382 1,2,3,6,12,20,24,60,120,120,360,720,2520,5040,20160,40320,181440,
%T A361382 362880,1814400,3628800,19958400,39916800,239500800,479001600,
%U A361382 3113510400,6227020800,43589145600,87178291200,653837184000,1307674368000,10461394944000,20922789888000
%N A361382 The orders, with repetition, of subset-transitive permutation groups.
%C A361382 If G is a permutation group on k letters, k > 0, then G induces a permutation of the subsets of size j for 0 <= j <= k. We call G subset-transitive if it has only one orbit of subsets for each j. G is subset-transitive if and only if it is (at least) floor(k/2)-transitive.
%C A361382 This restrictive condition admits only 1) symmetric groups of degree k for k >= 1, with order k! = A000142(k), which are k-transitive; 2) alternating groups of degree k for k >= 3, with order k!/2 = A001710(k), which are (k-2)-transitive; or 3) two exceptional groups, of orders 20 and 120.
%C A361382 The group of order 20 is AGL(1,5), which is 2-transitive on 5 letters.
%C A361382 The exceptional group of order 120 is PGL(2,5), which is 3-transitive on 6 letters, and contains AGL(1,5) as its one-point stabilizer. It is isomorphic as an abstract group, but not as a permutation group, to the symmetric group of degree 5. An outer automorphism of the symmetric group of degree 6 interchanges the two types of subgroup of order 120.
%H A361382 Hal M. Switkay, <a href="/A361382/b361382.txt">Table of n, a(n) for n = 1..48</a>
%H A361382 Shreeram S. Abhyankar, <a href="https://doi.org/10.1090/S0273-0979-1992-00270-7">Galois Theory on the Line in Non-Zero Characteristic</a>, Bulletin of the AMS, 27 (1992), 68-133.
%Y A361382 Cf. A000142, A001710, A187741.
%K A361382 nonn
%O A361382 1,2
%A A361382 _Hal M. Switkay_, Mar 09 2023