Automorphisms of Abelian group extensions. (English) Zbl 1209.20021
Let \(G\) be a group with an Abelian normal subgroup \(N\) and with factor group \(H\cong G/N\) – the authors are referring to this situation as to an Abelian extension.
Using cohomology techniques, the authors give special answers to the important questions of when can an automorphism of \(N\) (or, of \(G/N\)) be lifted to an automorphism of \(G\).
Applications are given, mostly for finite groups, in which case special such liftings are characterized by how the restrictions to the Sylow \(p\)-subgroups are behaving.
Using cohomology techniques, the authors give special answers to the important questions of when can an automorphism of \(N\) (or, of \(G/N\)) be lifted to an automorphism of \(G\).
Applications are given, mostly for finite groups, in which case special such liftings are characterized by how the restrictions to the Sylow \(p\)-subgroups are behaving.
Reviewer: Marian Deaconescu (Safat)
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20J06 | Cohomology of groups |
| 20E22 | Extensions, wreath products, and other compositions of groups |
| 20E36 | Automorphisms of infinite groups |
Keywords:
Abelian extensions; automorphisms of groups; split extensions; cohomology of groups; liftings of automorphismsReferences:
| [1] | Adney, J. E.; Yen, T., Automorphisms of \(p\)-groups, Illinois J. Math., 9, 137-143 (1965) · Zbl 0125.28803 |
| [2] | Brown, K. S., Cohomology of Groups (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0367.18012 |
| [3] | Buckley, J., Automorphism groups of isoclinic \(p\)-groups, J. Lond. Math. Soc., 12, 1, 37-44 (1975/1976) · Zbl 0358.20035 |
| [4] | Deaconescu, M.; Silberberg, G.; Walls, G. L., On commuting automorphisms of groups, Arch. Math., 79, 423-429 (2002) · Zbl 1017.20028 |
| [5] | Griess, R. L., Automorphisms of extraspecial groups and non-vanishing of degree 2 cohomology, Pacific J. Math., 48, 403-422 (1973) · Zbl 0283.20028 |
| [6] | Hall, P., The classification of prime power groups, J. Reine Angew. Math., 40, 130-141 (1940) · Zbl 0023.21001 |
| [7] | Jin, P., Automorphisms of groups, J. Algebra, 312, 562-569 (2007) · Zbl 1131.20037 |
| [8] | Robinson, D. J.S., Applications of cohomology to the theory of groups, (Groups — St. Andrews 1981. Groups — St. Andrews 1981, St. Andrews, 1981. Groups — St. Andrews 1981. Groups — St. Andrews 1981, St. Andrews, 1981, London Math. Soc. Lecture Note Ser., vol. 71 (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge-New York), 46-80 · Zbl 0496.20038 |
| [9] | Wells, C., Automorphisms of group extensions, Trans. Amer. Math. Soc., 155, 189-194 (1971) · Zbl 0221.20054 |
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.
