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Zbl 0603.20022
Glasby, S.P.
2-groups with every automorphism central. (English)
[J] J. Aust. Math. Soc., Ser. A 41, 233-236 (1986). ISSN 0263-6115

For $n\ge 3$ let $G\sb n$ be the 2-group generated by $x\sb 1,...,x\sb n$ with defining relations $x\sb i\sp{2\sp i}=1$, $[x\sb i,x\sb{i+1}]=x\sp{2\sp i}\sb{i+1}$ and all other commutators $[x\sb i,x\sb j]$ equal to 1. $G\sb n$ is not expressible as a non-trivial direct product (but the proof of this given in the paper is insufficient). All the automorphisms of $G\sb n$ are central; in fact the group of automorphisms of $G\sb n$ is non-Abelian of order $2\sp p$, where $p=(n-1)(2n\sp 2-n+6)/6$. Other examples of 2-groups with similar properties are given without proofs.
[N.Blackburn]

MSC 2000:: *20D45 Automorphisms of finite groups
20D15 Nilpotent finite groups

Keywords: central automorphisms; group of automorphisms; 2-groups

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