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Finite groups of outer automorphisms of free groups

Published online by Cambridge University Press:  18 May 2009

Bruno Zimmermann
Bruno Zimmermann
Affiliation:
Università Degli Studi di Trieste, Dipartimento di Scienze Matematiche, 34100 Trieste, Italy e-mail: [email protected]
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Let Fr denote the free group of rank r and Out Fr: = AutFr/Inn Fr the outer automorphism group of Fr (automorphisms modulo inner automorphisms). In [10] we determined the maximal order 2rr! (for r > 2) for finite subgroups of Out Fr as well as the finite subgroup of that order which, for r > 3, is unique up to conjugation. In the present paper we determine all maximal finite subgroups (that is not contained in a larger finite subgroup) of Out F3, up to conjugation (Theorem 2 in Section 3). Here the considered case r = 3 serves as a model case: our method can be applied for other small values of r (in principle for any value of r) but the computations become considerably longer and are more apt for a computer then; the method can also be applied to determine the maximal finite subgroups of the automorphism group Aut Fr of Fr. Note that the canonical projection Aut Fr ⃗ Out Fr is injective on finite subgroups of Aut Fr; however, not all finite subgroups of Out Fr lift to finite subgroups of Aut Fr.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 38 , Issue 3 , September 1996 , pp. 275 - 282
Copyright
Copyright © Glasgow Mathematical Journal Trust 1996

References

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