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16.62 (2006)

Solved

Let $G$ be a group such that every $\alpha \in \text{Aut } G$ fixes every conjugacy class of $G$ (setwise). Must $\text{Aut } G = \text{Inn } G$?

Contributor: D. MacHale

Progress

Not necessarily: the paper (H. Heineken, Arch. Math. (Basel), 33 (1979/80), no. 6, 497–503), in particular, produces finite $p$-groups all of whose automorphisms preserve all the conjugacy classes, while by Gaschütz' theorem every finite $p$-group has outer automorphisms (A. Mann, Letter of 20 April 2006).

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