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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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3
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0, 0, 0, 0, 0, 6, 0, 10, 0, 8, 0, 16, 0, 10, 0, 35, 0, 28, 0, 22, 10, 14, 0, 54, 0, 16, 19, 28, 0, 28, 0, 158, 0, 20, 0, 78, 0, 22, 16, 76, 0, 36, 0, 40, 0, 26, 0, 236, 0, 64, 0, 46, 0, 212, 14, 98, 22, 32, 0, 80, 0, 34, 36, 937, 0, 52, 0, 58, 0, 52, 0, 272
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OFFSET
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1,6
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COMMENTS
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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, Dec 03 2007
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LINKS
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FORMULA
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EXAMPLE
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a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.
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PROG
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(GAP) A083573 := function(n) local max, grp, i; max := 0; for i in [1..NumberSmallGroups(n)] do grp := SmallGroup(n, i); if (not IsAbelian(grp)) then max := Maximum(max, Sum(ConjugacyClassesSubgroups(grp), Size)); fi; od; return max; end; # Eric M. Schmidt, Sep 07 2012
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003
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EXTENSIONS
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STATUS
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approved
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