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A083573
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Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.
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2
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0, 0, 0, 0, 0, 6, 0, 10, 0, 8, 0, 16, 0, 10, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| A group G is non-Abelian iff there are two elements x,y such that xy != yx. Then <x> and <y> are nontrivial subgroups whose order divides the order of G which therefore cannot be prime (neither the square of a prime: there are only two nonisomorphic groups of that order which are both abelian; see A051532 for more). This also implies that a(n) >= 2+2+2 = 6 for all nonzero elements of this sequence and for even n=2m>4 there is the non-Abelian dihedral group D_m with A007503(m)=sigma(m)+tau(m)=A000005(m)+A000203(m), providing a lower bound. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 03 2007
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FORMULA
| a(n) = 0 <=> A060689(n)=0 <=> n is in A051532 ; otherwise a(n) >= 6 and a(2n) >= A007503(n). - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Dec 03 2007
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EXAMPLE
| a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.
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CROSSREFS
| Cf. A018216, A061034.
Cf. A051532, A060689, A007503.
Sequence in context: A153314 A019622 A187429 * A117006 A073764 A158897
Adjacent sequences: A083570 A083571 A083572 * A083574 A083575 A083576
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KEYWORD
| more,nonn,changed
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AUTHOR
| Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003
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