A059806 Minimal size of the center of G where G is a finite group of order n.

1, 2, 3, 4, 5, 1, 7, 2, 9, 1, 11, 1, 13, 1, 15, 2, 17, 1, 19, 1, 1, 1, 23, 1, 25, 1, 3, 2, 29, 1, 31, 2, 33, 1, 35, 1, 37, 1, 1, 2, 41, 1, 43, 2, 45, 1, 47, 1, 49, 1, 51, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 3, 2, 65, 1, 67, 1, 69, 1, 71, 1, 73, 1, 1, 2, 77, 1

Offset: 1

Author

Noam Katz (noamkj(AT)hotmail.com), Feb 24 2001

Keywords

• nonn

Comments

a(n) = n if and only if n belongs to sequence A051532 - Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

a(n) = 1 if and only if n occurs in A060702. - Eric M. Schmidt, Aug 27 2012

Examples

			a(6) = 1 because the symmetric group S_3 has trivial center.
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..2000
• MathOverflow, Center of p-groups

Crossrefs

Cf. A059807, A051532.

Programs

GAP

A059806 := function(n) local min, fact, i; if (n mod 6 = 0) then return 1; fi; if (IsPrimePowerInt(n)) then fact := Factors(n); if (Length(fact) <> 2) then return fact[1]; fi; fi; min := n; for i in [1..NumberSmallGroups(n)] do min := Minimum(min, Size(Center(SmallGroup(n, i)))); if (min = 1) then break; fi; od; return min; end; # Eric M. Schmidt, Aug 27 2012

Formula

For prime p and m >= 3, a(p^m) = p. - Eric M. Schmidt, Aug 27 2012

Extensions

More terms from Eric M. Schmidt, Aug 27 2012