From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: Groups with trivial center again Date: Fri, 20 Apr 2001 20:48:03 +0000 (UTC) Newsgroups: sci.math Summary: Orders for which all groups are nilpotent, or have nontrivial center In article <9bptum$ac4$1@agate.berkeley.edu>, magidin@math.berkeley.edu (Arturo Magidin) writes: >In article , >Ahmed Fares wrote: >>By the post of Derek Holt the sequence of numbers n such that there is >>a group of order n with trivial center begins with: >>1,6,10,12,14,18,20,21,22,24,26,... >> >>and this is (apart from the number 1) the same initial segment of >>sequence A056868 in the EIS: >> >>ID Number: A056868 >>Sequence: 6,10,12,14,18,20,21,22,24,26,28,30,34,36,38,39,40,42,44,46, >> >>48,50,52,54,55,56,57,58,60,62,63,66,68,70,72,74,75,76,78,80, >> 82,84,86,88,90,92,93,94,96,98,100 >>Name: Numbers which are not nilpotent numbers. >>Comments: A number is nilpotent if every group of order n is >>nilpotent. >>References J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. >> Monthly, 107 (Aug. 2000), 631-634. >>See also: Cf. A003277, A051532, A056866. Complement of A056867. >>Keywords: nonn,nice,easy,more >>Offset: 1 >>Author(s): njas, Sep 2, 2000 >> >> >>Are the two sequences (ignoring 1) identical ? > >Well, a nilpotent group always has nontrivial center (look at the >upper central series), which tells you that if there exists a group of >order n with trivial center, n cannot be a nilpotent number. Yes, but the converse is not true. In fact, the very next non-nilpotent number 28 on the list is not a nontrivial centre number. There is a non-nilpotent group of order 28, namely the dihedral group of that order, but every group of order 28 has nontrivial centre. (Proof: By Sylow's theorem a group G of order 28 has a normal Sylow 7-subgroup S of order 7. Since |Aut(S)| = 6, if T is a Sylow 2-subgroup of the group then |T|=4, so an element t of T of order 2 must centralize S, and then t is in the centre of G.) Derek Holt.