Character codegrees and element orders in symmetric and alternating groups (Q6601872)

Let \(G\) be a finite group and let \(\mathrm{Irr}(G)\) be the set of irreducible complex characters of \(G\). For \(\chi \in \mathrm{Irr}(G)\), the codegree of \(\chi\) in \(G\) is defined as \(\mathrm{cod}_{G}(\chi)= |G: \ker(\chi)|/\chi(1)\) [\textit{G. Qian} et al., J. Algebra 312, No. 2, 946--955 (2007; Zbl 1127.20009)]. If \(S\) is a finite set of \(\mathbb{N}\), then define \(\rho(S)=\prod_{s \in S}s\) and, for \(g \in G\), let \(\rho(g)=\rho(\pi(o(g)))\).\N\NThe main result in this paper is Theorem A: Let \(G\) be a symmetric or an alternating group. For any \(g \in G\), there exists \(\mathrm{Irr}(G)\) such that \(o(g)^{2} \cdot \rho(g)^{-1}\) divides \(\mathrm{cod}_{G}(\chi)\).\N\NThis answers in the affirmative to a conjecture by \textit{G. Qian} [Bull. Lond. Math. Soc. 53, No. 3, 820--824 (2021; Zbl 1493.20003)].