A178498 - OEIS

%I #29 Aug 29 2019 01:25:14

%S 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,

%T 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,

%U 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

%N Number of Frobenius groups of order n.

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

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

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

%C 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.]

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

%H James McCarron, <a href="https://math.stackexchange.com/questions/2936538/what-are-the-frobenius-groups-of-order-100">What are the Frobenius groups of order 100?</a>, Math StackExchange, 2018.

%H Bernard Schott and Jean-Louis Tu <a href="http://www.les-mathematiques.net/phorum/read.php?17,785127,785720#msg-785720">QDV8 & H62 : Hommage à Frobenius - Frobenius 8 - Exercice 8.2</a> (French mathematical forum les-mathematiques.net)

%H Jean-Pierre Serre <a href="https://arxiv.org/abs/math/0503154">Groupes finis</a>, ENS - 1978/1979; arXiv:math/0503154 [math.GR], 2005-2008 (in French).

%e a(18)=2, the two Frobenius groups of order 18 being

%e -> the dihedral group D_9 of order 18 and

%e -> 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.

%p GroupTheory:-NumFrobeniusGroups( n ) # _James McCarron_, Aug 28 2019

%Y Cf. A016825, A220211.

%K nonn

%O 1,18

%A _Jozsef Pelikan_, May 28 2010

%E a(100) corrected by _James McCarron_, Aug 28 2019