A178498 Number of Frobenius groups of order n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 2, 0, 3

Offset

1,18

Comments

In this sequence there are two infinite families of Frobenius groups:

1) The dihedral groups D_{2n+1} of order 2*(2n+1), that is, A016825 without 2.

2) The one-dimensional affine groups in the finite fields F_q, q >= 3, of order q(q-1) corresponding to A220211.

a(42)=2 and 42 is the smallest integer with a Frobenius group of each type: the dihedral group D_21 and the affine group in F_7. [Comments from Bernard Schott, Dec 21 2012.]

References

J. J. Rotman, An Introduction to the Theory of Groups (4th Edition), Springer-Verlag, pp. 254-256.

Links

Table of n, a(n) for n=1..100.

James McCarron, What are the Frobenius groups of order 100?, Math StackExchange, 2018.

Bernard Schott and Jean-Louis Tu QDV8 & H62 : Hommage à Frobenius - Frobenius 8 - Exercice 8.2 (French mathematical forum les-mathematiques.net)

Jean-Pierre Serre Groupes finis, ENS - 1978/1979; arXiv:math/0503154 [math.GR], 2005-2008 (in French).

Example

a(18)=2, the two Frobenius groups of order 18 being
-> the dihedral group D_9 of order 18 and
-> the semidirect product of Z(3)xZ(3) with Z(2), where Z(2) acts by mapping every element of Z(3)xZ(3) to its inverse.

Maple

GroupTheory:-NumFrobeniusGroups( n ) # James McCarron, Aug 28 2019

Crossrefs

Cf. A016825, A220211.

Sequence in context: A359763 A378006 A277017 * A353422 A095408 A357375

Adjacent sequences: A178495 A178496 A178497 * A178499 A178500 A178501

Keyword

nonn

Author

Jozsef Pelikan, May 28 2010

Extensions

a(100) corrected by James McCarron, Aug 28 2019

Status

approved