14.95 (1999)
Open(C. R. Leedham-Green, P. M. Neumann, J. Wiegold). For a finite $p$-group $P$, denote by $c = c(P)$ its nilpotency class and by $b = b(P)$ its breadth, that is, $p^b$ is the maximum size of a conjugacy class in $P$. Class-Breadth Problem: Is it true that $c \leqslant b + 1$ if $p \neq 2$?
Progress
So far, the best known bound is $c < \frac{p}{p-1}b + 1$ (C. R. Leedham-Green, P. M. Neumann, J. Wiegold, J. London Math. Soc. (2), 1 (1969), 409–420). For $p = 2$ for every $n \in \mathbb{N}$ there exists a 2-group $T_n$ such that $c(T_n) \geqslant b(T_n) + n$ (W. Felsch, J. Neubüser, W. Plesken, J. London Math. Soc. (2), 24 (1981), 113–122).
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