Automorphisms of a free group of infinite rank. (English) Zbl 1158.20009
St. Petersbg. Math. J. 19, No. 2, 215-223 (2008); translation from Algebra Anal. 19, No. 2, 74-85 (2007).
The authors study the automorphisms \(\operatorname{Aut}(F_\infty)\) of the free group \(F_\infty\) of infinite countable rank. They describe this group in terms of generators which arise naturally. They use the sets of upper triangular, lower triangular, permutational and column finite automorphisms with the obvious meaning. These sets form subgroups of \(\operatorname{Aut}(F_\infty)\). They define a convenient generalization of the above called strings in terms of which they describe \(\operatorname{Aut}(F_\infty)\). We omit here the detailed definitions of the above.
Their main results states that: Each automorphism in \(\operatorname{Aut}(F_\infty)\) is the composition of some IA-automorphism and some automorphism belonging to the subgroup generated by the lower triangular and the column-finite automorphisms.
They also give many theorems describing subgroups of \(\operatorname{Aut}(F_\infty)\), for example one of them states that: The group \(\operatorname{Aut}(F_\infty)\) contains two infinite countable families: one of maximal normal subgroups and the other of normal incomparable subgroups.
They finish by giving some results on the automorphism group of the infinitely generated relatively free group corresponding to some varieties of groups.
Their main results states that: Each automorphism in \(\operatorname{Aut}(F_\infty)\) is the composition of some IA-automorphism and some automorphism belonging to the subgroup generated by the lower triangular and the column-finite automorphisms.
They also give many theorems describing subgroups of \(\operatorname{Aut}(F_\infty)\), for example one of them states that: The group \(\operatorname{Aut}(F_\infty)\) contains two infinite countable families: one of maximal normal subgroups and the other of normal incomparable subgroups.
They finish by giving some results on the automorphism group of the infinitely generated relatively free group corresponding to some varieties of groups.
Reviewer: Stylianos Andreadakis (Athens)
MSC:
| 20E05 | Free nonabelian groups |
| 20F28 | Automorphism groups of groups |
| 20E36 | Automorphisms of infinite groups |
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