New lower bounds for the number of conjugacy classes in finite nilpotent groups

Document Type : Research Paper

Author

‎Department of‎ ‎Mathematics‎, ‎University of Hawaii‎, ‎Honolulu‎, ‎HI 96822‎, ‎USA

10.22108/ijgt.2021.128396.1687

Abstract

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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