From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: Sequence of groups starting from S_6 Date: Mon, 3 Dec 2001 09:54:51 +0000 (UTC) Newsgroups: sci.math Summary: Towers of automorphism groups terminate In article <66f84b63c429b94605905b4264a0ab30.22128@mygate.mailgate.org>, "Robin Chapman" writes: >"Yoav Gross" wrote in message >news:6n8fa6zn2p3e@legacy... > >> Define a sequence of groups by G_1=S_6 (symmetric group on 6 letters) >> and for n>=2 G_n=Aut(G_(n-1)) . >> Is this sequence becomes periodic eventually ? >> > >Isn't G_3 = G_2? Yes. G_3 = G_2 = Aut(A_6) = PGammaL(2,9). For any finite nonabelian simple group S, it is not hard to show that Aut(Aut(S)) =~ Aut(S). There is also an old result that says that for any group G_1 with trivial centre, the chain G_1 <= G_2 <= G_3 = ... (with above notation) eventually becomes constant. I seem to remember hearing (maybe it was on sci.math even!) that it had been proved more recently that for any finite group whatsoever, the sequence G_1, G_2, G_3 ... eventually becomes constant (or maybe periodic), but I don't remember the details. Derek Holt. ============================================================================== From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: Sequence of groups starting from S_6 Date: Tue, 4 Dec 2001 16:14:49 +0000 (UTC) Newsgroups: sci.math In article , olaveshta@my-deja.com (Ola) writes: >In article ><66f84b63c429b94605905b4264a0ab30.22128@mygate.mailgate.org>, > "Robin Chapman" writes: [see above --djr] >By comparing Aut(Aut(G)) and Aut(G) for dihedral groups of small >order they are the same but for the Dihedral group of order 16: >Aut(Aut(D_16)) has order 64 while Aut(D_16) has order 32 . > >However by the computations I made in GAP it looks like for "many" >Dihedral groups Aut(Aut(G)) and Aut(G) have the same order and maybe >they are even isomorphic . >Is there an "explanation" for that ? Well it is true for dihedral groups of twice odd order, because Aut(G) is then the full holomorph of a cyclic group of odd order, which is complete. (Note that these groups have trivial centre.) It is also true when for dihedral groups of four times odd order, and for the dihedral group of order 8. Apart from that I think it is false. Derek Holt. ============================================================================== From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: Sequence of groups starting from S_6 Date: Tue, 4 Dec 2001 14:14:41 +0000 (UTC) Newsgroups: sci.math In article , "Jim Heckman" writes: [deletia --djr] >> There is also an old result that says that for any group G_1 with trivial >> centre, the chain G_1 <= G_2 <= G_3 = ... (with above notation) >> eventually becomes constant. > >And this too? I can see why the G_i must be a chain (with each G_i >having trivial center), but not why they must become constant. This is a result of Wielandt, 1939. I know that there is a proof in Passman's book on Permutation groups, but unfortunately I do not have a copy to hand. >> I seem to remember hearing (maybe it was on sci.math even!) that it had been >> proved more recently that for any finite group whatsoever, the sequence >> G_1, G_2, G_3 ... eventually becomes constant (or maybe periodic), but I >> don't remember the details. > >I think I remember this from sci.math.research, and I'm pretty sure it was >periodic. What I was remembering was a result of Joel Hamkins (1998), but in fact he proves that the automorphism chain terminates only transfinitely. So there is no guarantee (according to his result anyway) that the chain will become periodic after finitely many steps. Derek Holt. ============================================================================== From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: Sequence of groups starting from S_6 Date: Wed, 5 Dec 2001 10:44:05 +0000 (UTC) Newsgroups: sci.math In article , "Jim Heckman" writes: > >On 4-Dec-2001, mareg@primrose.csv.warwick.ac.uk () wrote: > >> In article , >> "Jim Heckman" writes: >> [...] [G=G_1 is a group G_{i+1} = Aut(G_i)] >> >> I seem to remember hearing (maybe it was on sci.math even!) that it had been >> >> proved more recently that for any finite group whatsoever, the sequence >> >> G_1, G_2, G_3 ... eventually becomes constant (or maybe periodic), but I >> >> don't remember the details. >> > >> >I think I remember this from sci.math.research, and I'm pretty sure it was >> >periodic. >> >> What I was remembering was a result of Joel Hamkins (1998), but in fact >> he proves that the automorphism chain terminates only transfinitely. So >> there is no guarantee (according to his result anyway) that the chain will >> become periodic after finitely many steps. > >What is meant by "terminates [...] transfinitely"? Clearly the chain itself is >countable... There is a natural homomorphism G_i -> G_{i+1}, the inner automorphism map. The idea is that for a limit ordinal x, you define G_x as the direct limit of the system { G_i | i < x }, with the above homomorphisms. That way you can define G_i for all ordinals. Hamkins proved that for any starting group, there is an i such that G_i has trivial centre, and then an older result of Thomas shows that the series eventually becomes constant. For example, if G_1 = D_8, the dihedral group of order 8, then G_i =~ D_8 for all natural numbers i, and the maps G_i -> G_{i+1} all have kernel of order 2 and image of order 4, so you could regard the series as constant throughout. But the direct limit of this system is defined to be G_w which has order 2, and then G_{w+1} = 1, which of course has trivial centre. Derek Holt. ============================================================================== From: mareg@primrose.csv.warwick.ac.uk () Subject: Re: Sequence of groups starting from S_6 Date: Thu, 6 Dec 2001 15:06:18 +0000 (UTC) Newsgroups: sci.math In article , "Jim Heckman" writes: > >On 5-Dec-2001, mareg@primrose.csv.warwick.ac.uk () wrote: [partial quote of previous message deleted --djr] >Cool. How is the direct limit defined in general? Do the centers of the G_i >always eventually become constant as i approaches a limit ordinal? If you have a collection of groups G_i for any index set, and some homomorphisms \phi:G_i -> G_j, then you get the direct limit in general by taking the direct product of the G_i, and factoring out the normal closure of all g phi(g)^-1, for all maps phi and all g in G_i (where G_i is identified with the corresponding subgroup of the direct product). In the example, each G_i = < a, b | a^4 = b^2 = (ab)^2 = 1 > is mapped to G_{i+1} by a->a^2, b->b, and the direct limit is essentially the subgroup {1,b}, because a -> a^2 -> a^4 = 1 dies. In general, centres of each individual G_i map onto 1 in the direct limit, but that does nor mean that the direct limit has no centre. Derek Holt.