A converse for a theorem of Gallagher
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915610153189376 |
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| author | Chen, Xiaoyou Lewis, Mark L. |
| author_facet | Chen, Xiaoyou Lewis, Mark L. |
| contents | Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $χ\in {\rm Irr} (G)$ satisfies $χ_N$ is irreducible, then $χβ$ is irreducible and distinct for all $β\in {\rm Irr} (G/N)$. Furthermore, if $θ= χ_N$, then these are all of the irreducible constituents of $θ^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $π$-partial characters and that a partial converse of that theorem is true. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07383 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A converse for a theorem of Gallagher Chen, Xiaoyou Lewis, Mark L. Group Theory Primary 20C15, Secondary 20C20 Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $χ\in {\rm Irr} (G)$ satisfies $χ_N$ is irreducible, then $χβ$ is irreducible and distinct for all $β\in {\rm Irr} (G/N)$. Furthermore, if $θ= χ_N$, then these are all of the irreducible constituents of $θ^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $π$-partial characters and that a partial converse of that theorem is true. |
| title | A converse for a theorem of Gallagher |
| topic | Group Theory Primary 20C15, Secondary 20C20 |
| url | https://arxiv.org/abs/2511.07383 |