A note on I-automorphisms. (English) Zbl 0976.20022
An automorphism of a group \(G\) is said to be a power automorphism if it maps every subgroup of \(G\) onto itself. The set \(\text{PAut }G\) of all power automorphisms of \(G\) is an Abelian normal subgroup of the full automorphism group \(\operatorname{Aut} G\) of \(G\). An automorphism \(\alpha\) of \(G\) is called an I-automorphism if \(H^\alpha=H\) for each infinite subgroup \(H\) of \(G\). The structure of the group \(\text{IAut }G\) consisting of all I-automorphisms of the group \(G\) has been investigated by M. Curzio, S. Franciosi and the reviewer [Arch. Math. 54, No. 1, 4-13 (1990; Zbl 0664.20018)]. In this article the authors study groups \(G\) for which \(\text{IAut }G\neq\text{PAut }G\), proving that such groups must satisfy strong structural restrictions.
Reviewer: Francesco de Giovanni (Napoli)
Citations:
Zbl 0664.20018References:
[1] | Adjan, S. I., Periodic groups of odd exponent, Proceedings of the 2nd International Conference on the Theory of Groups, Canberra, 1973. Proceedings of the 2nd International Conference on the Theory of Groups, Canberra, 1973, Lecture Notes in Mathematics, 372 (1974), Springer-Verlag: Springer-Verlag Berlin, p. 8-12 · Zbl 0306.20044 |
[2] | Beidleman, J. C.; Dixon, M. R.; Robinson, D. J.S., The generalized Wielandt subgroup of a group, Canad. J. Math., 47, 246-261 (1995) · Zbl 0832.20047 |
[3] | Chernikov, S. N., Groups with given properties for systems of infinite subgroups, Dokl. Akad. Nauk SSSR, 171, 806-809 (1966) · Zbl 0183.02703 |
[4] | Cooper, C. D., Power automorphisms of a group, Math. Z., 107, 335-356 (1968) · Zbl 0169.33801 |
[5] | Curzio, M.; Franciosi, S.; de Giovanni, F., On automorphisms fixing infinite subgroups of groups, Arch. Math. (Basel), 54, 4-13 (1990) · Zbl 0664.20018 |
[6] | Kegel, O. H.; Wehrfritz, B. A.F., Locally Finite Groups (1973), North-Holland: North-Holland Amsterdam · Zbl 0195.03804 |
[7] | Robinson, D. J.S., A Course in the Theory of Groups (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0496.20038 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.