Characters and Sylow 2-subgroups of maximal class revisited
The representation theory of finite groups having Sylow \(2\)-subgroups of maximal class is a classical and widely studied topic. These 2-groups, which are exactly dihedral, semidihedral and generalized quaternion 2-groups, have been characterized from distinct points of views, and particularly, they coincide with the non-abelian 2-groups whose derived subgroup has index 4. In this paper, the authors provide two interesting ways to recognize from the character table whether a finite group \(G\) possesses a Sylow \(2\)-subgroup \(P\) of maximal class. Precisely, the main result establishes the equivalence among the following assertions: (a) \(|P/P'|=4\); (b) either \(|P|=4\) or there exists \(g\in P\) such that \(|G:{\mathbf C}_G(g)|\) is even and \({\mathbb Q}(\chi(g): \chi \in \text{Irr}(G))={\mathbb Q}(\zeta \pm \zeta^{-1})\), where \(\zeta\in {\mathbb C}\) is a primitive \(|P|/2\)-th root of unity; (c) the principal 2-block of \(G\) has exactly four irreducible complex characters of odd degree. The equivalence between properties (a) and (b) is quite elementary. However, the fact that (c) implies (a), which is a particular case of the Alperin-McKay conjecture (unsolved even for \(p=2\)), requires the classification of the finite simple groups. The authors also obtain a highly non-trivial characterization of the finite groups whose Sylow 3-subgroups have order 3. These are those groups whose principal 3-block possesses exactly three irreducible complex characters of degree not divisible by 3. The authors remark, however, that they have not been able to find a characterization of the fact that \(|P/P'|=p^2\) for primes \(p\geq 5\).
scientific article; zbMATH DE number 6881294
scientific article