A102842 Insipid numbers: n is defined to be insipid if "G is a primitive subgroup of the symmetric group S_n" implies that "G=A_n or G=S_n".

1, 2, 3, 4, 34, 39, 46, 51, 58, 69, 70, 75, 76, 86, 87, 88, 92, 93, 94, 95, 96, 99, 106, 111, 115, 116, 118, 123, 124, 134, 141, 142, 143, 145, 146, 147, 148, 154, 159, 160, 161, 166, 172, 177, 178, 184, 185, 187, 188, 189, 195, 201, 202, 204, 205, 206, 207, 209

Offset: 1

Author

David L. Harden, Feb 27 2005

Keywords

• nonn

Comments

A few basic properties: No prime p > 3 is in this sequence, since the subgroup of S_p generated by any p-cycle is primitive (and too small to be A_p or S_p when p>3).

It seems hard to find long gaps in this sequence. It seems plausible (this is implied by some conjectures in number theory) that there are infinitely many strings of 5 consecutive positive integers not in this sequence; however, I do not know of a construction which should yield infinitely many strings of 6 consecutive positive integers which are in the sequence (this may be just a reflection of my ignorance of the right families of finite groups); the largest example I know of a string of more than 5 consecutive integers not in this sequence has length 7 and first term 2^150-5.

If q is a power of a prime and d > 1 is a positive integer (except in the cases where d=2 and q <= 4, in which this construction yields symmetric or alternating groups), then (q^d-1)/(q-1) is not insipid for the following reason:

The group PGL(d,q) acts doubly transitively (and therefore primitively) on the (q^d-1)/(q-1) 1-dimensional subspaces of a d-dimensional vector space over the finite field of order q. In particular, when d=2, this number is q+1 and this is why each power of a prime (including the primes themselves) prevents the next positive integer from being insipid, in addition to being noninsipid itself. This is why the explanation for why 38 is noninsipid just said that 38=37+1.

The Magma code generates the insipid numbers <= U, with the exceptions of 1 and 2. Since I do not know Magma well enough to judge this for myself, it is possible that U has to be a constant (and not just another program variable) for this code to work properly.

This is the set of n such that n = 1 or 2 and A000019(n)=2.

The link gives all the insipid numbers < 1000, except for 1 and 2. - David L. Harden, Aug 15 2007

There are infinitely many insipid numbers. In fact, they are of density 1, because P. J. Cameron, P. M. Neumann and D. N. Teague proved that the number of non-insipid numbers less than n grows like 2n/log(n). - Sébastien Palcoux, Jul 23 2019

Examples

			39 is the next term after 34 because it is possible to construct primitive nonnormal subgroups of S_n for n=35,36,37 and 38:
35: 35=(7 3) and 3 < 7/2 so S_7 acts primitively on 35 points because S_7 has maximal subgroups isomorphic to S_3 x S_4.
36: 36=(9 2) and 2 < 9/2 so S_9 acts primitively on 36 points because S_9 has maximal subgroups isomorphic to S_2 x S_7.
37: 37 is prime.
38: 38=37+1.
		

References

• J. Dixon and B. Mortimer: Permutation groups. Springer 1996, 360pp.

Links

• David L. Harden, Table of n, a(n) for n = 1..486 (insipid numbers below 1000) [Corrected Aug 25 2007]
• P. J. Cameron, P. M. Neumann, D. N. Teague, On the degrees of primitive permutation groups, Math. Z. 180 (1982), 141-149.
• S. Palcoux, Are there infinitely many insipid numbers? (version: 2019-07-22), MathOverflow.
• C. M. Roney-Dougal, The primitive permutation groups of degree less than 2500, Journal of Algebra 292 (2005) 154-183.

Crossrefs

Cf. A000019.

Programs

Magma

[n : n in [1..U] | NumberOfPrimitiveGroups(n) eq 2];