Automorphism groups of free metabelian nilpotent groups.(English. Russian original)Zbl 0812.20019

Algebra Logic 29, No. 6, 480-483 (1990); translation from Algebra Logika 29, No. 6, 746-751 (1990).
Let $$\text{Aut}(M_{n,c})$$, $$n \geq 2$$, $$c \geq 2$$, be the automorphism group of the free $$n$$-generator metabelian nilpotent group $$M_{n,c}$$ of class $$c$$ with generators $$\{x_ 1, \dots,x_ n\}$$ and let $$T \leq \text{Aut} (M_{n,c})$$ be the subgroup consisting of all tame automorphisms of $$M_{n,c}$$. If $$c \leq 2$$ then $$T = \text{Aut}(M_{n,c})$$ whereas for $$c \geq 3$$, $$\text{Aut} (M_{n,c})$$ does contain non-tame automorphisms. Here the authors are interested in the problem of finding an economical generating set for $$\text{Aut}(M_{n,c})$$ in the general case $$n \geq 2$$, $$c \geq 3$$. Define $$\theta_ k$$, $$3 \leq k \leq c$$, by $$x_ 1 \to x_ 1[x_ 1, x_ 2, (k-2)x_ 1]$$, $$x_ i \to x_ i$$, $$i \neq 1$$, and set $$K = \langle T,\theta_ 3, \dots, \theta_ c \rangle$$. In this note is is proved that some power of every automorphism of $$M_{n,c}$$ lies in $$K$$.

MSC:

 20F28 Automorphism groups of groups 20E36 Automorphisms of infinite groups 20F18 Nilpotent groups 20F16 Solvable groups, supersolvable groups 20F05 Generators, relations, and presentations of groups
Full Text:

References:

 [1] S. Andreadakis, ”On the automorphisms of free groups and free nilpotent groups,” Proc. London Math. Soc.,15, 239–268 (1965). · Zbl 0135.04502 [2] S. Andreadakis, ”Generators for Aut(G), G free nilpotent,” Arch. Math.,42, 296–300 (1984). · Zbl 0533.20017 [3] S. Bachmuth, ”Induced automorphisms of free groups and free metabelian groups,” Trans. Am. Math. Soc.,122, 1–17 (1966). · Zbl 0133.28101 [4] R. M. Bryant and C. K. Gupta, ”Automorphism groups of free nilpotent groups,” Arch. Math.,52, 313–320 (1989). · Zbl 0638.20023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.