Even Character Degrees and Ito–Michler Theorem

Let \(\textrm{Irr}_2(G)\) ${\text{Irr}}_2(G)$ be the set of linear and even-degree irreducible characters of a finite group G. In this paper, we prove that G has a normal Sylow 2-subgroup if \(\sum \limits _{\chi \in \textrm{Irr}_2(G)} \chi (1)^m/\sum \limits _{\chi \in \textrm{Irr}_2(G)} \chi (1)^{m-1} < (1+2^{m-1})/(1+2^{m-2})\) $\sum _{\chi \in {\text{Irr}}_2(G)}\chi {(1)}^m/\sum _{\chi \in {\text{Irr}}_2(G)}\chi {(1)}^{m-1}<(1+2^{m-1})/(1+2^{m-2})$ for a positive integer m, which is the generalization of several recent results concerning the well-known Ito–Michler theorem.