Finite groups with large automizers for their nonabelian subgroups
For a subgroup \(H\) of a finite group \(G\) denote \(\Aut_G(H)=N_G(H)/C_G(H)\) to be the automizer of \(H\) in \(G\). Since \(\text{Inn}(H)\) is isomorphic to a subgroup of \(\Aut_G(H)\) which is embedded in \(\Aut(H)\), the authors identify \(\Aut_G(H)\) as small if \(\Aut_G(H)\cong\text{Inn}(H)\) and large if \(\Aut_G(H)\cong\Aut(H)\). H. Zassenhaus proved that a finite group \(G\) is abelian if and only if the automizers of all abelian subgroups are small. By replacing `small' with `large', this was supplemented by \textit{H. Bechtell}, \textit{G. Silberberg} and \textit{M. Deaconescu} [Can. Math. Bull. 40, No. 3, 266-270 (1997; Zbl 0891.20016)] and extended to all subgroups by \textit{J. Lennox} and \textit{J. Wiegold} [Rend. Semin. Mat. Univ. Padova 89, 83-86 (1993; Zbl 0802.20026)]. In this article, the authors prove that the collection of finite groups with large automizers for their nonabelian subgroups comprises those which are holomorphs of groups of odd prime order.
scientific article; zbMATH DE number 1145409