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<p>Motivated by Malle and Navarro’s work on characterizing normal Sylow <i>p</i>-subgroups by character degrees, we investigate whether the normality of a Hall subgroup <i>H</i> of a finite group <i>G</i> can be characterized by the degrees of irreducible constituents of the induced character (1<sub><i>H</i></sub>)<sup><i>G</i></sup>, where 1<sub><i>H</i></sub> is the trivial character of <i>H</i>. We prove that this characterization is valid for solvable groups and that the general case can be reduced to the study of non-abelian simple groups. Furthermore, we show that for solvable groups, the normality of a Hall subgroup can be detected solely by the monomial constituents of the induced character.</p>

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Characterizing normal Hall subgroups by character degrees

  • 期刊论文

  • 发表日期：

    2026年7月1日

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  • Yang Liu,
  • Yong Yang,
  • Jiping Zhang

摘要

Motivated by Malle and Navarro’s work on characterizing normal Sylow p-subgroups by character degrees, we investigate whether the normality of a Hall subgroup H of a finite group G can be characterized by the degrees of irreducible constituents of the induced character (1H)G, where 1H is the trivial character of H. We prove that this characterization is valid for solvable groups and that the general case can be reduced to the study of non-abelian simple groups. Furthermore, we show that for solvable groups, the normality of a Hall subgroup can be detected solely by the monomial constituents of the induced character.