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

1, 2, 3, 6, 12, 20, 24, 60, 120, 120, 360, 720, 2520, 5040, 20160, 40320, 181440, 362880, 1814400, 3628800, 19958400, 39916800, 239500800, 479001600, 3113510400, 6227020800, 43589145600, 87178291200, 653837184000, 1307674368000, 10461394944000, 20922789888000

Offset: 1

Author

Hal M. Switkay, Mar 09 2023

Keywords

• nonn

Comments

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.

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.

The group of order 20 is AGL(1,5), which is 2-transitive on 5 letters.

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.

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..48
• Shreeram S. Abhyankar, Galois Theory on the Line in Non-Zero Characteristic, Bulletin of the AMS, 27 (1992), 68-133.

Crossrefs

Cf. A000142, A001710, A187741.