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A361382
The orders, with repetition, of subset-transitive permutation groups.
2
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
1,2
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
Shreeram S. Abhyankar, Galois Theory on the Line in Non-Zero Characteristic, Bulletin of the AMS, 27 (1992), 68-133.
CROSSREFS
KEYWORD
nonn
AUTHOR
Hal M. Switkay, Mar 09 2023
STATUS
approved