?Which group theoretic properties of a permutation group are inherited by its closure

‎‎Given a permutation group $G$ on a finite set $\Omega$‎, ‎for $k\geq 2$‎, ‎the $k$-closure $G^{(k)}$ of $G$ is the largest subgroup of $\Sym(\Omega)$ with the same orbits as $G$ on the set $\Omega^k$ of $k$-tuples from $\Omega$‎. Then $G\leq\ldots\leq G^{(k)}\leq G^{(k-1)}\leq\ldots\leq G^{(2)}$‎. ‎In this paper‎, ‎we review some basic properties of $k$-closures of permutation groups and report some group theoretic properties of a permutation group which are inherited by its closures‎. ‎We prove that‎‎ G$ is abelian of exponent $e$ if and only if $G^{(2)}$ is abelian of exponent $e$‎$ G$ is a $p$-group‎, ‎$p$ a prime‎, ‎if and only if $G^{(2)}$ is a $p$-group‎$ G$ is nilpotent if and only if $G^{(2)}$ is nilpotent‎$