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On automorphisms of some \(p\)-groups. (English) Zbl 1062.20021

Summary: A group \(G\) is said to enjoy the ‘Hasse princple’ if every local coboundary of \(G\) is a global coboundary. Let \(G\) be a nonabelian finite \(p\)-group of order \(p^m\), \(p\) prime and \(m>4\), having a normal cyclic subgroup of order \(p^{m-2}\) but having no element of order \(p^{m-1}\). We prove that \(G\) enjoys the ‘Hasse principle’ if \(p\) is odd, but in the case \(p=2\), there are fourteen such groups, twelve of which enjoy the ‘Hasse principle’ but the remaining two do not satisfy it. We also find all the conjugacy preserving outer automorphisms for these two groups.

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D45 Automorphisms of abstract finite groups
20J05 Homological methods in group theory
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References:

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