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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A characterization of periodic automorphisms of a free group


Author: James McCool
Journal: Trans. Amer. Math. Soc. 260 (1980), 309-318
MSC: Primary 20E05; Secondary 20E36
DOI: https://doi.org/10.1090/S0002-9947-1980-0570792-1
MathSciNet review: 570792
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Abstract: Let $ \theta $ be an automorphism of finite order of a free group X. We characterise the action of $ \theta $ on X by showing that X has a free basis which is the disjoint union of finite subsets $ {S_j}$, where if $ {S_j}\, = \,\{ {u_0},\,{u_1},\, \ldots ,\,{u_k}\} $ then $ {u_i}\theta \, = \,{u_{i\, + \,1}}\,(0\, \leqslant \,1\, < \,k)$ and $ {u_k}\theta \, = \,{A_j}u_0^e{B_j}$ for some $ {A_j}$, $ {B_j}$ in X and $ \varepsilon \, = \, \pm \,1$. As an application of this result, we obtain a list of the conjugacy classes of periodic automorphisms of the free group of rank three.


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DOI: https://doi.org/10.1090/S0002-9947-1980-0570792-1
Article copyright: © Copyright 1980 American Mathematical Society