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Finitely generated groups with virtually free automorphism groups

Published online by Cambridge University Press:  20 January 2009

Martin R. Pettet
Martin R. Pettet
Affiliation:
Department of Mathematiics, University of Toledo, Toledo, Ohio 43606, U.S.A.
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Abstract

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It is shown that the full automorphism group of a finitely generated group G is virtually free if and only if the center Z(G) is finitely generated of torsion-free rank r at most two and, depending on the value of r, the central quotient G/Z(G) belongs to one of three precisely defined classes of virtually free groups. Some consequences and special cases are also discussed.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 3 , October 1995 , pp. 475 - 484
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Alperin, J. L., Groups with finitely many automorphisms, Pacific J. Math. 12 (1962), 1–5.CrossRefGoogle Scholar
2.Baer, R., Finite extensions of abelian groups with minimum conditions, Trans. Amer. Math. Soc. 79 (1955), 521–540.CrossRefGoogle Scholar
3.Collins, D. J., The automorphism group of a free product of finite groups, Arch. Math. (Basel) 50 (1988), 385–390.CrossRefGoogle Scholar
4.Dyer, J. L., A remark on automorphism groups, Contemp. Mat. 33 (1984), 208–211.CrossRefGoogle Scholar
5.Alajdzievski, S. K., Automorphism group of a free group: centralizers and stabilizers, J. Algebra 150 (1992), 435–502.CrossRefGoogle Scholar
6.Karrass, A., Pietrowski, A. and Solitar, D., Automorphisms of a free product with an amalgamated subgroup, Contemp. Math. 33 328–340.CrossRefGoogle Scholar
7.Krstic, S., Finitely generated virtually free groups have finitely presented automorphism groups, Proc. London Math. Soc. (3) 64 (1992), 49–69.CrossRefGoogle Scholar
8.Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory (Springer-Verlag, Berlin, 1977).Google Scholar
9.Mccool, J., The automorphism groups of finite extensions of free groups, Bull. London Math. Soc. 20 (1988), 131–135.CrossRefGoogle Scholar
10.Miller, G. A., On the multiple holomorphs of a group, Math. Annale. 66 (1908), 133–142.CrossRefGoogle Scholar
11.Pettet, M. R., The automorphism group of a virtual direct product of groups, in Proceedings of the A.M.S. Special Session, Infinite Groups and Group Rings, Tuscaloosa 1992 (World Scientific, Singapore, New Jersey, London, Hong Kong, 1993), 102–110.Google Scholar
12.Robinson, D. J. S., A Course in the Theory of Groups (Springer-Verlag, New York, 1982).CrossRefGoogle Scholar
13.Serre, J.-P., Trees (Springer-Verlag, Berlin, 1980).CrossRefGoogle Scholar