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Outer automorphisms of hypercentral p-groups

  • Orazio Puglisi (a1)

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In his celebrated paper [3] Gaschiitz proved that any finite non-cyclic p-group always admits non-inner automorphisms of order a power of p. In particular this implies that, if G is a finite nilpotent group of order bigger than 2, then Out (G) = Aut(G)/Inn(G) =≠1. Here, as usual, we denote by Aut (G) the full group of automorphisms of G while Inn (G) stands for the group of inner automorphisms, that is automorphisms induced by conjugation by elements of G. After Gaschiitz proved this result, the following question was raised: “if G is an infinite nilpotent group, is it always true that Out (G)≠1?”

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References

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1.Buckley, J. and Wiegold, J., On the number of outer automorphisms of an infinite nilpotent p-groups, Arch. Math. (Basel) 31 (1978), 321328.
2.Dixon, M. R., personal communication.
3.Gaschütz, W., Nichtabelsche p-Gruppen besitzen aussere p-Automorphismen, J. Algebra 4 (1966), 12.
4.Menegazzo, F. and Stonehewer, S., On the automorphism group of a nilpotent p-group, J. London Math. Soc. 31 (1985), 272276.
5.Puglisi, O., On outer automorphisms of Cernikov p-groups, Rend. Sem. Mat. Univ. Padova 83 (1990), 97106.
6.Puglisi, O., A note on the automorphism group of a locally finite p-group, Bull. London Math. Soc. 24 (1992), 437441.
7.Thomas, S., Complete existentially closed locally finite groups, Arch. Math. (Basel) 44 (1985), 97109.
8.Zalesskiĩ, A. E., An example of torsion-free nilpotent group having no outer automorphisms, Mat. Zametki 11 (1972), 221226 English translation, Math. Notes, 11(1972), 16–19.
9.Zalesskiĩ, A. E., A nilpotent p-group has outer automorphisms, Dokl. Akad. Nauk. SSSR 196 (1971); English translation, Soviet Math. Doklady, 12 (1971), 227230.

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Outer automorphisms of hypercentral p-groups

  • Orazio Puglisi (a1)

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