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.

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

Offset

1,6

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, Dec 03 2007

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..511

Formula

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

Example

a(6)=6 because the only non-Abelian group with 6 elements is S_3 with 6 subgroups.

Prog

(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

Crossrefs

Cf. A018216, A061034.

Cf. A051532, A060689, A007503.

Sequence in context: A153314 A019622 A187429 * A117006 A294345 A073764

Adjacent sequences: A083570 A083571 A083572 * A083574 A083575 A083576

Keyword

nonn

Author

Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003

Extensions

More terms from Eric M. Schmidt, Sep 07 2012

Status

approved