14.89 (1999)

(E. A. O’Brien, A. Shalev). Let $P$ be a finite $p$-group of order $p^m$ and let $m = 2n + e$ with $e = 0$ or 1. By a theorem of P. Hall the number of conjugacy classes of $P$ has the form $n(p^2 - 1) + p^e + a(p^2 - 1)(p - 1)$ for some integer $a \geqslant 0$, which is called the abundance of $P$.
$\qquad$ a) Is there a bound for the coclass of $P$ which depends only on $a$? Note that $a = 0$ implies coclass 1 and that all known examples with $a = 1$ have coclass $\leqslant 3$. (The group $P$ has coclass $r$ if $\lvert P \rvert = p^{c+r}$ where $c$ is the nilpotency class of $P$.)
$\qquad$ b) Is there an element $s \in P$ such that $\lvert C_P(s) \rvert \leqslant p^{f(a)}$ for some $f(a)$ depending only on $a$? We already know that we can take $f(0) = 2$ and it seems that $f(1) = 3$.

Note that A. Jaikin-Zapirain (J. Group Theory, 3, no. 3 (2000), 225–231) has proved that $\lvert P \rvert \leqslant p^{f(p,a)}$ for some function $f$ of $p$ and $a$ only.