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

Author

Jozsef Pelikan, May 28 2010

Keywords

• nonn

Comments

From Bernard Schott, Dec 21 2012: (Start)

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. (End)

Examples

			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.
		

References

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

Links

• 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).

Crossrefs

Cf. A016825, A220211.

Programs

Maple

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

Extensions

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