Automorphisms of torsion-free nilpotent groups of class two
HTML articles powered by AMS MathViewer
- by Manfred Dugas and Rüdiger Göbel PDF
- Trans. Amer. Math. Soc. 332 (1992), 633-646 Request permission
Abstract:
We construct $2$-divisible, torsion-free abelian groups $G$ admitting an alternating bilinear map. We use these groups $G$ to find nilpotent groups $N$ of class $2$ such that $\operatorname {Aut}(N)$ modulo a natural normal subgroup is a prescribed group.References
- Reinhold Baer, Groups with abelian central quotient group, Trans. Amer. Math. Soc. 44 (1938), no. 3, 357–386. MR 1501972, DOI 10.1090/S0002-9947-1938-1501972-1 A. L. S. Corner, Groups of units of orders in ${\mathbf {Q}}$-algebras, Abelian Group Theory, Proc. Caribbean Conf. 1991 (L. Fuchs and R. Göbel, eds.), (to appear).
- A. L. S. Corner and Rüdiger Göbel, Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), no. 3, 447–479. MR 779399, DOI 10.1112/plms/s3-50.3.447
- Manfred Dugas and Rüdiger Göbel, Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 45 (1982), no. 2, 319–336. MR 670040, DOI 10.1112/plms/s3-45.2.319 —, Solution of Philip Hall’s problem on the existence of complete locally finite $p$-groups and results on $ULF$ groups, submitted 1990.
- Manfred Dugas and Rüdiger Göbel, Torsion-free nilpotent groups and $E$-modules, Arch. Math. (Basel) 54 (1990), no. 4, 340–351. MR 1042126, DOI 10.1007/BF01189580
- Manfred Dugas and Rüdiger Göbel, Outer automorphisms of groups, Illinois J. Math. 35 (1991), no. 1, 27–46. MR 1076664
- Manfred Dugas and Rüdiger Göbel, All infinite groups are Galois groups over any field, Trans. Amer. Math. Soc. 304 (1987), no. 1, 355–384. MR 906820, DOI 10.1090/S0002-9947-1987-0906820-7
- P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801. MR 105441
- P. Hall and B. Hartley, The stability group of a series of subgroups, Proc. London Math. Soc. (3) 16 (1966), 1–39. MR 191946, DOI 10.1112/plms/s3-16.1.1 H. Heineken, Automorphisms groups of torsion-free nilpotent groups of class two, Symposia Math. 17 (1976), 235-250.
- Hermann Heineken and Hans Liebeck, The occurrence of finite groups in the automorphism group of nilpotent groups of class $2$, Arch. Math. (Basel) 25 (1974), 8–16. MR 349844, DOI 10.1007/BF01238631
- Sadayoshi Kojima, Isometry transformations of hyperbolic $3$-manifolds, Topology Appl. 29 (1988), no. 3, 297–307. MR 953960, DOI 10.1016/0166-8641(88)90027-2
- Michel Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190 (French). MR 0088496 W. J. LeVeque, Topics in number theory, Vol. I, Addison-Wesley, Reading, Mass., 1961.
- Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879
- Takao Matumoto, Any group is represented by an outerautomorphism group, Hiroshima Math. J. 19 (1989), no. 1, 209–219. MR 1009671
- Martin R. Pettet, On inner automorphisms of finite groups, Proc. Amer. Math. Soc. 106 (1989), no. 1, 87–90. MR 968625, DOI 10.1090/S0002-9939-1989-0968625-8 —, Characterizing inner automorphisms of groups, manuscript.
- Paul E. Schupp, A characterization of inner automorphisms, Proc. Amer. Math. Soc. 101 (1987), no. 2, 226–228. MR 902532, DOI 10.1090/S0002-9939-1987-0902532-X
- Saharon Shelah, A combinatorial principle and endomorphism rings. I. On $p$-groups, Israel J. Math. 49 (1984), no. 1-3, 239–257. MR 788269, DOI 10.1007/BF02760650
- Robert B. Warfield Jr., Nilpotent groups, Lecture Notes in Mathematics, Vol. 513, Springer-Verlag, Berlin-New York, 1976. MR 0409661
- U. H. M. Webb, The occurrence of groups as automorphisms of nilpotent $p$-groups, Arch. Math. (Basel) 37 (1981), no. 6, 481–498. MR 646507, DOI 10.1007/BF01234386
- A. E. Zalesskiĭ, An example of a nilpotent group without torsion that has no outer automorphisms, Mat. Zametki 11 (1972), 21–26 (Russian). MR 291291
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 332 (1992), 633-646
- MSC: Primary 20F29
- DOI: https://doi.org/10.1090/S0002-9947-1992-1052906-0
- MathSciNet review: 1052906