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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.11199 |
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| _version_ | 1866918011694219264 |
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| author | Çınarcı, Burcu Keller, Thomas Michael Maróti, Attila Simion, Iulian I. |
| author_facet | Çınarcı, Burcu Keller, Thomas Michael Maróti, Attila Simion, Iulian I. |
| contents | In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of conjugacy classes of nontrivial $p$-elements in $G$. Using a recent theorem of Giudici, Morgan and Praeger, we prove that there exists a function $f(x)$ with $f(x) \to \infty$ as $x \to \infty$ such that $n(G) \geq f(|G|)$ for any finite group $G$. Let $G$ be a finite group, and let $p$ be a prime dividing $|G|$. Let $k_{p'}(G)$ denote the number of conjugacy classes of elements of $G$ whose orders are coprime to $p$. We show that either $p=11$ and $G=C_{11}^2\rtimes \text{\rm SL}(2,5)$, or there exists a factorization $p-1 = ab$ with $a$ and $b$ positive integers, such that $k_{p}(G) \geq a$ and $k_{p'}(G) \geq b$ with equalities in both cases if and only if $G=C_p \rtimes C_b$ with $C_G(C_p) = C_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | More on Landau's theorem and Conjugacy Classes Çınarcı, Burcu Keller, Thomas Michael Maróti, Attila Simion, Iulian I. Group Theory In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of conjugacy classes of nontrivial $p$-elements in $G$. Using a recent theorem of Giudici, Morgan and Praeger, we prove that there exists a function $f(x)$ with $f(x) \to \infty$ as $x \to \infty$ such that $n(G) \geq f(|G|)$ for any finite group $G$. Let $G$ be a finite group, and let $p$ be a prime dividing $|G|$. Let $k_{p'}(G)$ denote the number of conjugacy classes of elements of $G$ whose orders are coprime to $p$. We show that either $p=11$ and $G=C_{11}^2\rtimes \text{\rm SL}(2,5)$, or there exists a factorization $p-1 = ab$ with $a$ and $b$ positive integers, such that $k_{p}(G) \geq a$ and $k_{p'}(G) \geq b$ with equalities in both cases if and only if $G=C_p \rtimes C_b$ with $C_G(C_p) = C_p$. |
| title | More on Landau's theorem and Conjugacy Classes |
| topic | Group Theory |
| url | https://arxiv.org/abs/2406.11199 |