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Automorphisms of \(p\)-groups of maximal class. (English) Zbl 1156.20020

Summary: A. Juhász [Trans. Am. Math. Soc. 270, 469-481 (1982; Zbl 0488.20023)] has proved that the automorphism group of a group \(G\) of maximal class of order \(p^n\), with \(p\geq 5\) and \(n>p+1\), has order divisible by \(p^{\lceil(3n-2p+5)/2\rceil}\).
We show that by translating the problem in terms of derivations, the result can be deduced from the case where \(G\) is metabelian. Here one can use a general result of A. Caranti and C. M. Scoppola [Arch. Math. 56, No. 3, 218-227 (1991; Zbl 0693.20038)] concerning automorphisms of two-generator, nilpotent metabelian groups.

MSC:

20D45 Automorphisms of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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References:

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