On automorphisms permuting generators in groups of rank two |
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| Author |
WTOMASZ@zeus.polsl.gliwice.pl
Institute of Mathematics,Silesian Technical University,Kaszubska 23,44-100 Gliwice,POLAND
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| Abstract |
Let $F$ be a free group on two generators, and let
$\sigma$ be the automorphism of $F$ permuting the
generators. Let $N$ be a $\sigma$-invariant, normal subgroup of
$F$, and let $\bar{\sigma}$ be the automorphism of $F/N$ induced by
$\sigma$.
Let $G$ be a group of rank two, and let $\alpha$ be its
automorphism of order two. A question we consider is: when is
$\alpha$ induced by $\sigma$? We give some examples of
groups in which every automorphism of order two is induced by
$\sigma$ and examples of groups with an automorphism of order two
which is not induced by $\sigma$.
We also give an~example of a group which has two
non-conjugated automorphisms, both induced by~$\sigma$.
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| Keywords |
free group, generators, automorphism, σ-invariant normal subgroup, induced automorphism, rank two group, automorphism of order two, examples, non-conjugated automorphisms
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| DOI |
doi:10.30970/ms.8.2.207-211
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Reference |
1. H.S.M. Coxeter, W.O.J. Moser Generators and relations for discrete groups, Springer-Verlag,addr Berlin, Heidelberg, New York– 1972
2. E.I. Khukhro Nilpotent Groups and their Automorphisms, Walter de Gruyte,addr Berlin, New York–1993 3. G. Higman Groups and rings which have automorphisms without non-trivial fixed elements// J. London Math. Soc. V.50–1854 p.8–15 4. O. Macedońska, D. Solitar On binary $\sigma$-invariant words in a group// Contemporary Math. V.169--1994 p.431--449 5. B.H. Neumann Groups with automorphisms that leave only the neutral element fixed// Arch. Math. (Basel) V.7 –1956 p.1–5 6. E. Płonka Symmetric words in nilpotent groups of class $\leq$ 3// Fundamenta Math. V. 97--1977 p.95--103 |
| Pages |
207-211
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| Volume |
8
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| Issue |
2
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| Year |
1997
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| Journal |
Matematychni Studii
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| Full text of paper | |
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