A character theoretic criterion for a \(p\)-closed group.
By a result of \textit{S. M. Gagola} jun. and \textit{M. L. Lewis} [Commun. Algebra 27, No. 3, 1053-1056 (1999; Zbl 0929.20010)] a finite group \(G\) is nilpotent if and only if \(\chi(1)^2\) divides \(|G/\ker\chi|\) for any irreducible character \(\chi\) of \(G\). Now let \(p\) be a prime, and for an integer \(n\) denote by \(n_p\) the largest power of \(p\) dividing \(n\). The author obtains the following \(p\)-local version (and thus generalization) of the Gagola-Lewis result. Suppose that \(|G/\ker\chi|_p\geq(\chi(1)^2)_p\) for all \(\chi\in\text{Irr}(G)\). Then one of the following holds: (1) \(G\) has a normal Sylow \(p\)-subgroup; (2) \(p=3\), and \(G\) has a composition factor \(A_7\); (3) \(p=2\), and \(G\) has a composition factor belonging to \(\{A_7,A_{11},A_{13},M_{22}\}\). As an immediate consequence of this result, a finite \(p\)-solvable group \(G\) has a normal Sylow \(p\)-subgroup if and only if \(|G/\ker\chi|_p\geq(\chi(1)^2)_p\) for all \(\chi\in\text{Irr}(G)\). A couple of corollaries are also proved, for example, the finite group \(G\) is 3-closed if and only if \(|G/\ker\chi|_3>(\chi(1)^2)_3\) for every \(\chi\in\text{Irr}(G)\) whose degree is divisible by 3.
scientific article; zbMATH DE number 6105517
scientific article