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A presentation of the automorphism group of the two-generator free metabelian and nilpotent group of class \(c\). (English) Zbl 1008.20026

Ukr. Math. J. 54, No. 6, 945-956 (2002) and Ukr. Mat. Zh. 54, No. 6, 771-779 (2002).
The authors give a presentation of the group \(\operatorname{Aut}(G)\), where \(G=\text{gp}\{x,y\}=M_{2,c}\) is a free metabelian nilpotent group of class \(c\). The main result of the paper is the following: \(\text{IA}(G)/\text{Inn}(G)\) is a free Abelian group of rank \((c-2)(c+3)/2\) and freely generated by \(\xi_{ij}\), \(\chi_k\), \(0\leq i,j\leq c-2\), \(1\leq i+j\leq c-2\), \(1\leq k\leq c-2\), where \(\xi_{ij}=[x\to x[x,y]^{(x-1)^i(y-1)^j}\), \(y\to y]\) and \(\chi_k=[x\to x\), \(y\to y[x,y]^{(x-1)^k}]\). Here \(\text{Inn}(G)=G/\zeta(G)\simeq M_{2,c-1}\) is a free metabelian nilpotent group of class \(c-1\).

MSC:

20F28 Automorphism groups of groups
20F05 Generators, relations, and presentations of groups
20F16 Solvable groups, supersolvable groups
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