04118588

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04118588 Kassimatis, Nick Bass and Serre theory and Nielsen transformations (free and free product case). Bull. Greek Math. Soc. 27, 39-46 (1986). 1986 Greek Mathematical Society, Athens EN G-trees transformations on trees Nielsen transformations graph isomorphism Zbl 0427.20016 Zbl 0665.20001 Let G be a group, and let $T\sb 1$, $T\sb 2$ be G-trees with vertex sets $V\sb 1$, $V\sb 2$ and edge sets $E\sb 1$, $E\sb 2$ respectively. W. Dicks showed [see {\it W. Dicks}, Groups, Trees and Projective Modules (Lect. Notes Math. 790, 1980; Zbl 0427.20016) p. 59-60; or {\it W. Dicks} and {\it M. J. Dunwoody}, Groups acting on Graphs (Camb. Stud. Adv. Math. 17, 1989; Zbl 0665.20001) p. 91] that if there is a G-isomorphism $V\sb 1\to V\sb 2$ then there is a G-isomorphism $E\sb 1\to E\sb 2$. However, these isomorphisms will not in general combine to give a graph isomorphism. In this paper the author defines certain transformations on trees (``Nielsen transformations of type N''). He then shows the following. Suppose that G acts freely on $E\sb 1$, and that the quotient graphs $G\setminus T\sb 1$, $G\setminus T\sb 2$ are finite. Given a G- isomorphism $V\sb 1\to V\sb 2$ there is a tree $T\sb 2'$, with vertex set $V\sb 2$, such that the given map $V\sb 1\to V\sb 2$ extends to a graph isomorphism $T\sb 1\to T\sb 2'$. S.J.Pride