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New lower bounds for the number of conjugacy classes in finite nilpotent groups | ||
| International Journal of Group Theory | ||
| دوره 11، شماره 2، شهریور 2022، صفحه 109-119 اصل مقاله (417.07 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2021.128396.1687 | ||
| نویسنده | ||
| Edward A. Bertram* | ||
| Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA | ||
| چکیده | ||
| 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 $(c,t)$. | ||
| کلیدواژهها | ||
| Nilpotent؛ conjugacy؛ derived series | ||
| مراجع | ||
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