13.21 (1995)

a) Is there an infinite finitely generated residually finite $p$-group, in which the order $|g|$ of an arbitrary element $g$ does not exceed $f(\delta(g))$, where $\delta(g)$ is the length of $g$ with respect to a fixed set of generators and $f(n)$ is a function growing at $n \to \infty$ slower than any power function $n^\lambda, \lambda > 0$?
b) What is the minimal possible rate of growth of the function $\pi(n) = \max_{\delta(g)\leqslant n} |g|$ for the class of groups indicated in part a) of this problem? It is known (R. I. Grigorchuk, Math. USSR–Izv., 25 (1985), 259–300) that there exist $p$-groups with $\pi(n) \leqslant n^\lambda$ for some $\lambda > 0$. At the same time, it follows from a result of E. I. Zel’manov that $\pi(n)$ is not bounded if $G$ is infinite.