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 $(c,t)$.
Published Version
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