-----BEGIN PGP SIGNED MESSAGE----- Hash: SHA1 Hello, all!
[I realize this is off-topic. I posted it to sci.math.research, but the response rate there is sometimes poor, and I know you are all experts on group theory. Please excuse the off-topic post.] Recall that a group is perfect if and only if its abelianization is trivial. Equivalently, P is perfect if and only if its first homology group, H_1(P) = 0. A groups S is superperfect if and only if H_1(S) = H_2(S) = 0. A finitely presented group is superperfect if it admits a "balanced presentation", that is, a presentation with the same number of generators and relators. The fundamental group of a homology sphere (a manifold with the same integral homology of a sphere) is always superperfect. The automorphism group of a group is all of its automorphisms. The inner automorphisms are a normal subgroup of the automorphism group, and the outer automorphism group of a group is its automorphism group mod its inner automorphism group. Does anyone know of a finitely presented superperfect group (or, more preferably, and (explicitly defined) infinite family of f.p. superperfect groups) with infinite outer automorphism group? It would be helpful if you could explicitly present the outer automorphism group (or otherwise explicitly define it). I only really have experience with one superperfect group, namely SL(2,5), the special linear group in dimension two over the field with 5 elements. It's outer automorphism group is Z_2, about as far as you can get from infinite. Any assistance you can provide would be appreciated. Sincerely, - -- Jeffrey Rolland <rolla...@uwm.edu> -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.10-svn4923 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org iQEVAwUBSlZVig7x0haAVywNAQKBiwf5Ad0l7CITG3OrhfLu5+XrWkc+zo0widVF 0aD/B+XnFBkpvAUpb6HObyr5GVj1yW9wFPxN57Tg3bOsm+vWiFHvNHx3ZH4IsxwZ FsY8ETD/s8TEg3sFWMx4IQ4Mlq8rN7tlp6Mlv7aPhzbQZ2bAICmqrlzMEOp24uph Btstb2gMkgvJOWgDzFl5AXC13rRb8RaiV+xWidoRNjPyE63cfy3gbsCznL3xxtGF RdLMHe01TQaKH9ovmM1HG9qAtzcCram//B/WQuCPeAjNObSg3CqIfu4jIL1SG9Vc Wq+CnPT69Fh6O8q3WoRruefYEAkahh3OBmwpemYsQpswnLyV04njXg== =h4xx -----END PGP SIGNATURE----- _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
