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A note on the existence of noninner automorphisms of order \(p\) in some finite \(p\)-groups. (English) Zbl 1475.20036

By a celebrated theorem of Gaschütz, every finite nonabelian \(p\)-group has a noninner automorphism of \(p\)-power order. Berkovich conjectured a stronger statement: every finite nonabelian \(p\)-group admits a noninner automorphism of order \(p.\) The authors prove this conjecture in the particular case when \(p\) is odd and \(|Z_3(G)/Z(G)| \leq p^{d(G)+1},\) where \(Z_3(G)\) is the third member of the upper central series of \(G\) and \(d(G)\) is the minimal number of generators of \(G.\) As a consequence, every finite \(p\)-group of order less than \(p^9\) has a noninner automorphism of order \(p.\)

MSC:

20D45 Automorphisms of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups

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References:

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