On non self-normalizing subgroups
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913587610517504 |
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| author | Bianchi, Mariagrazia Camina, Rachel D. Lewis, Mark L. Pacifici, Emanuele Sanus, Lucia |
| author_facet | Bianchi, Mariagrazia Camina, Rachel D. Lewis, Mark L. Pacifici, Emanuele Sanus, Lucia |
| contents | Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and its interplay with solvability and nilpotency. We first show that if $G$ belongs to $D_n$ with $n \le 3$, then $G$ is solvable of derived length at most 2. We also show that $A_5$ is the unique nonsolvable group in $D_4$, and that $SL_2(3)$ is the unique solvable group in $D_4$ whose derived length is larger than 2. For a group $G$, we define $D(G)$ to be the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. We determine the relationship between $D(H \times K)$ and $D(H)$ and $D(K)$. We show that if $G$ is nilpotent and lies in $D_n$, then $G$ has nilpotency class at most $n/2$ and its derived length is at most $\log_2 (n/2) + 1$. We consider $D_n$ for several classes of Frobenius groups, and we use this classification to classify the groups in $D_0$, $D_1$, $D_2$, and $D_3$. Finally, we show that if $G$ is solvable and lies in $D_n$ with $n \ge 3$, then $G$ has derived length at most the minimum of $n-1$ and $3 \log_2 (n+1) + 9$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_18102 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On non self-normalizing subgroups Bianchi, Mariagrazia Camina, Rachel D. Lewis, Mark L. Pacifici, Emanuele Sanus, Lucia Group Theory Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and its interplay with solvability and nilpotency. We first show that if $G$ belongs to $D_n$ with $n \le 3$, then $G$ is solvable of derived length at most 2. We also show that $A_5$ is the unique nonsolvable group in $D_4$, and that $SL_2(3)$ is the unique solvable group in $D_4$ whose derived length is larger than 2. For a group $G$, we define $D(G)$ to be the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. We determine the relationship between $D(H \times K)$ and $D(H)$ and $D(K)$. We show that if $G$ is nilpotent and lies in $D_n$, then $G$ has nilpotency class at most $n/2$ and its derived length is at most $\log_2 (n/2) + 1$. We consider $D_n$ for several classes of Frobenius groups, and we use this classification to classify the groups in $D_0$, $D_1$, $D_2$, and $D_3$. Finally, we show that if $G$ is solvable and lies in $D_n$ with $n \ge 3$, then $G$ has derived length at most the minimum of $n-1$ and $3 \log_2 (n+1) + 9$. |
| title | On non self-normalizing subgroups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2411.18102 |