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On \(p\)-groups with abelian automorphism group. (English) Zbl 0829.20028

In 1913 Miller has shown that there exists a group of order \(2^6\) whose automorphism group is abelian. Here it is shown, that for an odd prime \(p\) no finite non-cyclic \(p\)-group of order \(p^6\) with abelian automorphism group exists. On the other hand, an example of a finite \(p\)- group of order \(p^7\) is given whose automorphism group is abelian. This group is special, is generated by 4 elements, and is the smallest of an infinite family of groups of order \(p^{n^2+3n+3}\), where \(n\) is a natural number.
Reviewer: B.Amberg (Mainz)

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D45 Automorphisms of abstract finite groups
20F28 Automorphism groups of groups
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References:

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