A059829 Maximal size of a minimal-generating-set of G where G is a finite group of order n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 1, 4, 1, 3, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 2, 1, 5, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 2, 3, 1, 2, 1, 4, 2, 3, 2, 2, 1, 2, 1, 2, 2, 6, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2

Offset

1,4

Comments

a(n) <= floor(log_2(n)) with equality if n=2^m is a power of 2.

For n >= 2, a(n) = 1 iff n belongs to sequence A003277.

a(n) >= A051903(n). - Álvar Ibeas, Mar 28 2015

a(p^2) = 2 for all primes p, since there are only two groups (up to isomorphism) of order p^2: Z_p^2 and Z_p X Z_p. The latter is generated by 2 elements. - Tom Edgar, Apr 06 2015

Links

Jinyuan Wang, Table of n, a(n) for n = 1..511 (terms 1..150 from Robert Israel)

Example

Up to isomorphism, there are five groups of order 8: the two non-abelian groups (the dihedral group and the quaternion group) are both generated by two elements, and the three abelian groups are Z_8 (generated by 1 element), Z_2 X Z_4 (generated by 2 elements) and Z_2 X Z_2 X Z_2 (generated by 3 elements). Thus a(8) = 3.

Prog

(GAP)
A := [0];
for n in [2 .. 100] do
   G := AllSmallGroups(n);
   m := NumberSmallGroups(n);
   t := 1;
   for i in [ 1 .. m] do
     while EulerianFunction(G[i], t) = 0 do
       t:= t+1;
     od;
   od;
   A[n]:= t;
od;
A; # Robert Israel, Apr 01 2015

Crossrefs

Cf. A003277, A051903.

Sequence in context: A256106 A077480 A327524 * A363369 A304465 A304687

Adjacent sequences: A059826 A059827 A059828 * A059830 A059831 A059832

Keyword

nonn

Author

Noam Katz (noamkj(AT)hotmail.com), Feb 25 2001

Extensions

Offset and first term corrected by Álvar Ibeas, Mar 27 2015

More terms from Robert Israel, Apr 01 2015

Status

approved