A220211 The order of the one-dimensional affine group in the finite fields F_q with q >= 3.

6, 12, 20, 42, 56, 72, 110, 156, 240, 272, 342, 506, 600, 702, 812, 930, 992, 1332, 1640, 1806, 2162, 2352, 2756, 3422, 3660, 4032, 4422, 4970, 5256, 6162, 6480, 6806, 7832, 9312, 10100, 10506, 11342, 11772, 12656, 14520, 15500, 16002, 16256, 17030, 18632

Offset: 1

Author

Bernard Schott, Dec 07 2012

Keywords

• nonn
• easy

Comments

The affine group is the group of invertible affine transformations in F_q such as: x--> ax+b, a > 0.

These groups are Frobenius groups belonging to A178498.

F_q is a field, so q = p^n, p is prime, with q >= 3 here.

The one-dimensional affine group in the finite fields F_q with q >= 3 is isomorphic to the semidirect product F_q x F_q^{*}, where F_q is endowed with the law +, and F_q^{*} is endowed with the law x. - Bernard Schott, Dec 22 2012

Examples

			a(1)=6 and this affine group of order 6 in the field F_3 is the dihedral group D_3 isomorphic to permutation group S_3.
a(2)=12 and this affine group of order 12 in the field F_4 is the semidirect product of Z(2) X Z(2) with Z(3).
a(6)=72 because for p=3, n=2 ==> q = p^n = 9 and 72 = q(q-1) = 9*8.
		

Links

• 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.

Crossrefs

Cf. A000961, A178498.

Programs

Maple

(p,n)-> p^n*(p^n-1)

Mathematica

mx = 20000; t = {}; p = 2; While[cnt = 0; n = 1; While[m = p^n (p^n - 1); m <= mx, AppendTo[t, m]; cnt++; n++]; cnt > 0, p = NextPrime[p]]; Union[Rest[t]] (* T. D. Noe, Dec 19 2012 *)

Formula

For the finite field F_q with q = p^n, the order of its affine group is q(q-1) = p^n(p^n-1), p prime, q >= 3.

a(n) = A000961(n+2)*(A000961(n+2) - 1). - Bernard Schott, Dec 22 2012