An $S_k$-set in a group $\Gamma $ is a set $A\subseteq \Gamma $ such that $\alpha _1\dots \alpha _k=\beta _1\dots \beta _k$ with $\alpha _i,\beta _i\in A$ implies $(\alpha _1,\ldots ,\alpha _k)=(\beta _1,\ldots ,\beta _k)$. An $S_k'$-set is a set such that $\alpha _1\beta _1^{-1}\dots \alpha _k\beta _k^{-1}=1$ implies that there exists i such that $\alpha _i=\beta _i\text { or }\beta _i=\alpha _{i+1}$. We give explicit constructions of large $S_k$-sets in the groups $\mathrm {Sym}(n)$ and $\mathrm {Alt}(n)$ and $S_2$-sets in $\mathrm {Sym}(n)\times \mathrm {Sym}(n)$ and $\mathrm {Alt}(n)\times \mathrm {Alt}(n)$. We give probabilistic constructions which yield large $S_2'$-sets in $\mathrm {Sym}(n)$. We also give upper bounds on the size of $S_k$-sets in certain groups, improving the trivial bound by a constant multiplicative factor. We describe some connections between $S_k$-sets and extremal graph theory. In particular, we determine up to a constant factor the minimum outdegree of a digraph which guarantees even cycles with certain orientations. As applications, we improve the upper bound on Hamilton paths which pairwise create a two-part cycle of given length, and we show that a directed version of the Erdős–Simonovits compactness conjecture is false.