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 (۳/۲) \log_۲ |G| when G is nilpotent‎. ‎Using only Hall's theorem‎, ‎this is the best one can do when |G| = ۲^n‎. ‎Using a result of G.J‎. ‎Sherman‎, ‎we improve the constant ۳/۲ to ۵/۳‎, ‎which is best possible across all nilpotent groups and to ۱۵/۸ when G is nilpotent and |G| \ne ۸,۱۶‎. ‎These results are then used to prove that k(G) > \log_۳(|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)‎.

کلیدواژه ها:

‎Nilpotent‎ ، ‎conjugacy‎ ، ‎derived series