Character degrees and local subgroups revisited
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912318946803712 |
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| author | Martínez, J. Miquel |
| author_facet | Martínez, J. Miquel |
| contents | Let $p$ and $q$ be different primes and let $G$ be a finite $q$-solvable group. We prove that $\mathrm{Irr}_{p'}(G)\subseteq \mathrm{Irr}_{q'}(G)$ if and only if $\mathbf{N}_G(P)\subseteq \mathbf{N}_G(Q)$ and $\mathbf{C}_{Q'}(P)=1$ for some $P\in\mathrm{Syl}_p(G)$ and $Q\in\mathrm{Syl}_q(G)$. Further, if $B$ is a $q$-block of $G$ and $p$ does not divide the degree of any character in $\mathrm{Irr}(B)$ then we prove that a Sylow $p$-subgroup of $G$ is normalized by a defect group of $B$. This removes the $p$-solvability condition of two theorems of G. Navarro and T. R. Wolf. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Character degrees and local subgroups revisited Martínez, J. Miquel Group Theory Representation Theory 20C20, 20C15 Let $p$ and $q$ be different primes and let $G$ be a finite $q$-solvable group. We prove that $\mathrm{Irr}_{p'}(G)\subseteq \mathrm{Irr}_{q'}(G)$ if and only if $\mathbf{N}_G(P)\subseteq \mathbf{N}_G(Q)$ and $\mathbf{C}_{Q'}(P)=1$ for some $P\in\mathrm{Syl}_p(G)$ and $Q\in\mathrm{Syl}_q(G)$. Further, if $B$ is a $q$-block of $G$ and $p$ does not divide the degree of any character in $\mathrm{Irr}(B)$ then we prove that a Sylow $p$-subgroup of $G$ is normalized by a defect group of $B$. This removes the $p$-solvability condition of two theorems of G. Navarro and T. R. Wolf. |
| title | Character degrees and local subgroups revisited |
| topic | Group Theory Representation Theory 20C20, 20C15 |
| url | https://arxiv.org/abs/2411.15968 |