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| Médium: | Preprint |
| Vydáno: |
2024
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| Témata: | |
| On-line přístup: | https://arxiv.org/abs/2402.06708 |
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| _version_ | 1866913230807367680 |
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| author | Beltrán, Antonio Felipe, María José Melchor, Carmen |
| author_facet | Beltrán, Antonio Felipe, María José Melchor, Carmen |
| contents | Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there exists only a finite number of finite groups $G$, up to isomorphism, having a normal subgroup $N$ of index $n$ which contains exactly $s$ non-central $G$-conjugacy classes. We provide upper bounds for the orders of $G$ and $N$, which are used by using GAP to classify all finite groups with normal subgroups having a small index and few $G$-classes. We also study the corresponding problems when we only take into account the set of $G$-classes of prime-power order elements contained in a normal subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Landau's Theorem on conjugacy classes for normal subgroups Beltrán, Antonio Felipe, María José Melchor, Carmen Group Theory Landau's theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes for any positive integer $k$. We show that, for any positive integers $n$ and $s$, there exists only a finite number of finite groups $G$, up to isomorphism, having a normal subgroup $N$ of index $n$ which contains exactly $s$ non-central $G$-conjugacy classes. We provide upper bounds for the orders of $G$ and $N$, which are used by using GAP to classify all finite groups with normal subgroups having a small index and few $G$-classes. We also study the corresponding problems when we only take into account the set of $G$-classes of prime-power order elements contained in a normal subgroup. |
| title | Landau's Theorem on conjugacy classes for normal subgroups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2402.06708 |