\(p\)-groups of maximal class as automorphism groups
An interesting problem in group theory is that of deciding which groups \(H\) are the full automorphism group of some group \(G\) (i.e. when is the equation \(\Aut G\cong H\) soluble?). Many examples are known of groups \(H\) for which there is no such \(G\), the easiest being the case when \(H\) is a finite non-trivial cyclic group of odd order. Several papers have been concerned with showing that groups from some specific class cannot be the full automorphism group of a group. This paper makes a further contribution to this discussion. The authors prove the following remarkable theorem. Let \(p\) be a prime and \(A\) a finite \(p\)-group of maximal class. Then there is a group \(G\) such that \(\Aut G\cong A\) if and only if one of the following cases occur: (1) \(p=2\) and \(A\cong D_8\) (when \(G\cong D_8\) or \(G\cong C_4\times C_2\)) or \(A\cong Q_8\) (when \(G\) is torsion-free Abelian) or (2) \(p=3\) and there is an integer \(n>1\) such that \(A\cong\langle x,y,t\mid x^{3^n}=y^{3^n}=1,\;[x,y]=t^3=x^{3^{n-1}},\;x^t=x^{-2}y^{-3},\;y^t=xy\rangle\), in which case \(G\) is infinite of nilpotency class 3. In each case the nilpotency class of \(A\) is even and \(G'\) is cyclic. The interested reader should also consult \textit{T. Fournelle} [J. Algebra 70, 16-22 (1981; Zbl 0457.20040) and J. Algebra 80, 106-112 (1983; Zbl 0509.20040)] and \textit{U. Martin} [Bull. Am. Math. Soc., New Ser. 15, 78-82 (1986; Zbl 0599.20032)].
scientific article; zbMATH DE number 1993624