  The spectrum of a finite group is the set of its element orders. We give an
affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that
for every positive integer $k$, the $k$-th direct power of the simple linear
group $L_{n}(2)$ is uniquely determined by its spectrum in the class of finite
groups provided $n$ is a power of $2$ greater than or equal to $56k^2$.
