  A group law is said to be detectable in power subgroups if, for all coprime
$m$ and $n$, a group $G$ satisfies the law if and only if the power subgroups
$G^m$ and $G^n$ both satisfy the law. We prove that for all positive integers
$c$, nilpotency of class at most $c$ is detectable in power subgroups, as is
the $k$-Engel law for $k$ at most 4. In contrast, detectability in power
subgroups fails for solvability of given derived length: we construct a finite
group $W$ such that $W^2$ and $W^3$ are metabelian but $W$ has derived length
$3$. We analyse the complexity of the detectability of commutativity in power
subgroups, in terms of finite presentations that encode a proof of the result.
