A060702 Orders of finite groups that have trivial center.

1, 6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 30, 34, 36, 38, 39, 42, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 66, 68, 70, 72, 74, 75, 78, 80, 82, 84, 86, 90, 93, 94, 96, 98, 100, 102, 106, 108, 110, 111, 114, 116, 118, 120, 122, 126, 129, 130, 132, 134, 136

Offset: 1

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001

Keywords

• nonn

Comments

Apart from the first element 1 this is a subsequence of A056868 because a nilpotent group has nontrivial center. If n = 0 mod 6 or n >= 6 and n = 2 mod 4 then n is in this sequence.

If n >= 6 and n == 2 mod 4 then n is a member of the sequence because of the dihedral group of order 2(2k+1). In addition, if p is a prime and p == 1 mod 4 then n=4p is a member of the sequence; however, if p == 3 mod 4 and p>5, then n=4p is not a member of the sequence. Furthermore, if n=pq where p and q are distinct odd primes with pDes MacHale and Mossie Crowe, Jul 05 2005

This sequence is closed under multiplication. - Eric M. Schmidt, Aug 27 2012

Examples

			The symmetric group S_3 of order 6 has trivial center so 6 belongs to the sequence.
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A056868, A059806.

For the corresponding numbers of centerless groups of these orders see A357900.

Extensions

The old entry 89 was an error, since it is a prime. - Robert F. Bailey (robertb(AT)math.carleton.ca) and Brett Stevens (brett(AT)math.carleton.ca), Jul 16 2009

Sequence extended and corrected by Eric M. Schmidt, Aug 27 2012