Bounding the number of p'-degrees from below

Thomas Keller; Yong Yang

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Bounding the number of p'-degrees from below

Thomas Keller, Yong Yang

Group Theory · Submitted 20 Nov 2023

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove a recent conjecture by Hung stating that $|\Irr_{p'}(G)|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that $p^a$ does not exceed the exponent of the center of $P$.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,120 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.23 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in\Syl_p(G)$. We prove a recent conjecture by Hung stating that $|\Irr_{p'}(G)|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that $p^a$ does not exceed the exponent of the center of $P$. We then also show that the number of conjugacy classes of elements of $G$ for which $p^a$ is the exact $p$-part of their order is at least $p^{a-1}$.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in\Syl_p(G)$.	background 1.00	0.07	0.01	asserted 0.54
s1	We prove a recent conjecture by Hung stating that $\|\Irr_{p'}(G)\|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that $p^a$ does not exceed the exponent of the center of $P$.	result 0.79	0.91	0.89	asserted 1.00
s2	We then also show that the number of conjugacy classes of elements of $G$ for which $p^a$ is the exact $p$-part of their order is at least $p^{a-1}$.	result 0.98	0.95	0.13	asserted 1.00

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Original abstract

Let $G$ be a finite group of order divisible by a prime $p$ and let $P\in\Syl_p(G)$. We prove a recent conjecture by Hung stating that $|\Irr_{p'}(G)|\geq \frac{\exp(P/P')-1}{p-1}+2\sqrt{p-1}-1.$ Let $a\geq 2$ be an integer and suppose that $p^a$ does not exceed the exponent of the center of $P$. We then also show that the number of conjugacy classes of elements of $G$ for which $p^a$ is the exact $p$-part of their order is at least $p^{a-1}$.

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