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 A220211 The order of the one-dimensional affine group in the finite fields F_q with q >= 3. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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] 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. 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] EXAMPLE 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. 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 *) CROSSREFS Cf. A000961, A178498. Sequence in context: A117343 A286290 A240521 * A028611 A220470 A141808 Adjacent sequences:  A220208 A220209 A220210 * A220212 A220213 A220214 KEYWORD nonn AUTHOR Bernard Schott, Dec 07 2012 STATUS approved

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Last modified September 20 06:59 EDT 2021. Contains 347577 sequences. (Running on oeis4.)