New lower bounds for the number of conjugacy classes in finite nilpotent groups
P. Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) \ge (3/2) \log_2 |G|$ when $G$ is nilpotent. Using only Hall's theorem, this is the best one can do when $|G| = 2^n$. Using a result of G.J. Sherman, we improve the constant $3/2$ to $5/...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2022-06-01
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| Series: | International Journal of Group Theory |
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| Online Access: | https://ijgt.ui.ac.ir/article_25810_6cf96773007f3bab56c3708c2139b4e7.pdf |
| Summary: | P. Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) \ge (3/2) \log_2 |G|$ when $G$ is nilpotent. Using only Hall's theorem, this is the best one can do when $|G| = 2^n$. Using a result of G.J. Sherman, we improve the constant $3/2$ to $5/3$, which is best possible across all nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| \ne 8,16$. These results are then used to prove that $k(G) > \log_3(|G|)$ when $G/N$ is nilpotent, under natural conditions on $N \trianglelefteq G$. Also, when $G'$ is nilpotent of class $c$, we prove that $k(G) \ge (\log |G|)^t$ when $|G|$ is large enough, depending only on $(c,t)$. |
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| ISSN: | 2251-7650 2251-7669 |