Shotgun identification on groups

We consider the problem of shotgun identification patterns on groups, which extends previous work DNA sequences and labeled graphs. A a group $G$ is specified by two finite subsets $C \subset G$ $K alphabet $\mathcal{A}$. In such problems, there ``global" pattern $w \in \mathcal{A}^{CK}$, one would like to be able identify this (up translation) based only observation ``local" $K$-shaped subpatterns $w$, called reads, centered at elements $C$. an asymptotic regime in size $w$ tends infinity symbols are drawn i.i.d. fashion. Our first general result establishes sufficient conditions under random identifiable from its reads with probability tending one, our second non-identifiable one. Additionally, we illustrate main results applying them several families examples.