Exponent of a finite group of odd order with an involutory automorphism (Q2311087)

Let \(G\) be a finite group of odd order, \(\phi \in \mathrm{Aut}(G)\) an involutory automorphism, \(G_{\phi}=C_{G}(\phi)\) and \(G_{-\phi}=\{g \in G \mid g^{\phi}=g^{-1}\}\). The paper under review contains some partial generalizations of a result obtained by the reviewer et al. [Rend. Semin. Mat. Univ. Padova 129, 1--15 (2013; Zbl 1300.20031)] if \(G\) is a \(p\)-group. In particular, Theorem 1.1 asserts that if for every \(x,y \in G_{-\phi}\) we have \(x^{e}=1\), the derived length of \(\langle x,y \rangle\) is at most \(d\) and if \(G_{\phi}\) is nilpotent of class \(c\), then the exponent of \([G, \phi]=\langle G_{-\phi} \rangle\) is bounded by a function of \(d\), \(e\) and \(c\).