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

1,2

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 p<q, then pq belongs to the sequence if and only if p divides q-1. - Des MacHale and Mossie Crowe, Jul 05 2005

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

Links

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

Example

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

Crossrefs

Cf. A056868, A059806.

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

Sequence in context: A161543 A056868 A069209 * A054741 A177712 A176552

Adjacent sequences: A060699 A060700 A060701 * A060703 A060704 A060705

Keyword

nonn

Author

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

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

Status

approved