20.58 (2022)

Let $\omega(G)$ denote the set of element orders of a finite group $G$. A finite group $G$ is said to be recognizable (by spectrum) if every finite group $H$ with $\omega(H) = \omega(G)$ is isomorphic to $G$.
$\qquad$ (a) Is it true that for every $n$ there is a recognizable group that is the $n$-th direct power of a nonabelian simple group?
$\qquad$ (b) Is it true that there is a nonabelian simple group $L$ such that for every $n$ there is a recognizable group whose socle is the $k$-th direct power of $L$ for some $k \geqslant n$?