An $S_k$-set in a group $Γ$ is a set $A\subseteqΓ$ such that $α_1\cdotsα_k=β_1\cdotsβ_k$ with $α_i,β_i\in A$ implies $(α_1,\ldots,α_k)=(β_1,\ldots,β_k)$. An $S_k'$-set is a set such that $α_1β_1^{-1}\cdotsα_kβ_k^{-1}=1$ implies that there exists $i$ such that $α_i=β_i\text{ or }β_i=α_{i+1}$. We give explicit constructions of large $S_k$-sets in the group $S_n$ and $S_2$-sets in $S_n\times S_n$ and $A_n\times A_n$. We give probabilistic constructions for `nice' groups which obtain large $S_2$-sets in $A_n$ and $S_2'$-sets in $S_n$. 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.