A generalized character associated to element orders (Q6607866)

Let \(G\) be a finite group, and let \(o(g)\) denote the order of an element \(g \in G\). The author proves that \(\vert G\vert \) is the smallest positive integer \(m\) such that the class function \(g \mapsto m\, o(g)\) is a generalized character of \(G\). Moreover, the generalized character \(\Xi\) of \(G\), defined by \(\Xi(g) := \vert G\vert \, o(g)\) for \(g \in G\), is an integral linear combination of permutation characters \({1_C}^G\) where \(C\) ranges over the cyclic subgroups of \(G\). The author also gives a formula for the multiplicity of an irreducible character \(\chi\) of \(G\) in \(\Xi\). When \(\chi\) is the trivial character of \(G\) then this multiplicity is equal to \(\psi(G) := \sum_{g \in G} o(g)\), an invariant frequently studied in group theory.