cp's OEIS Frontend

A083573 Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.

%I A083573 #11 Sep 07 2012 17:54:20
%S A083573 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,
%T A083573 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,
%U A083573 14,98,22,32,0,80,0,34,36,937,0,52,0,58,0,52,0,272
%N A083573 Maximal number of subgroups in a non-Abelian group with n elements, or zero if there are no non-Abelian groups of order n.
%C A083573 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
%H A083573 Eric M. Schmidt, <a href="/A083573/b083573.txt">Table of n, a(n) for n = 1..511</a>
%F A083573 a(n) = 0 <=> A060689(n)=0 <=> n is in A051532 ; otherwise a(n) >= 6 and a(2n) >= A007503(n). - _M. F. Hasler_, Dec 03 2007
%e A083573 a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.
%o A083573 (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
%Y A083573 Cf. A018216, A061034.
%Y A083573 Cf. A051532, A060689, A007503.
%K A083573 nonn
%O A083573 1,6
%A A083573 Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003
%E A083573 More terms from _Eric M. Schmidt_, Sep 07 2012