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Finite \(p\)-groups with small automorphism group. (English) Zbl 1319.20019

The paper under review disproves the well-known conjecture, that the order of a nonabelian, finite \(p\)-group divides the order of its automorphism group. Various special cases of the conjecture had been solved in the affirmative in several papers over the years. We refer to the introduction of the paper for a list of these contributions.
For each prime \(p\), the authors construct a family \((U_i)\) of finite \(p\)-groups whose orders tend to infinity, but such that \(\limsup_{i\to\infty}|\operatorname{Aut}(U_i)|/|U_i|^{40/41}<\infty\). This entails the existence of groups that disprove the conjecture. The construction relies on exhibiting first an infinite, finitely generated pro-\(p\) group \(U\) such that its automorphism group is in some sense smaller than \(U\). Here \(U\) is taken to be a uniform \(p\)-adic pro-\(p\) group, so that one can use the dimension as a measure of size. The authors then write \(U\) as the inverse limit of suitable finite \(p\)-groups \(U_i\), and use a cohomological approach to show that these groups satisfy the above requirements.

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D45 Automorphisms of abstract finite groups
20E18 Limits, profinite groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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