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A178498
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Number of Frobenius groups of order n.
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1
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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
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OFFSET
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1,18
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COMMENTS
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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.]
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REFERENCES
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J. J. Rotman, An Introduction to the Theory of Groups (4th Edition), Springer-Verlag, pp. 254-256.
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LINKS
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Jean-Pierre Serre Groupes finis, ENS - 1978/1979; arXiv:math/0503154 [math.GR], 2005-2008 (in French).
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EXAMPLE
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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.
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MAPLE
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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