On the subgroup perfect codes in Cayley graphs (Q2678977)

A subset \(C\) of the vertex set of a graph \(G\) is called a perfect \(t\)-code for \(t\) greater than or equal to 1 if every vertex of \(G\) is at distance not exceeding \(t\) to exactly one vertex of \(C\). A perfect 1-code is called a perfect code. A perfect code can also be thought of as an efficient dominating set or an independent perfect dominating set. It is customary to look at perfect codes in Cayley graphs to extend the notion of perfect codes in graphs. It also acts as a generalization of Hamming codes and Lee codes. An interesting open problem is the following: is it true that \(H\) is a perfect code of \(G\) if and only if a Sylow 2-subgroup of \(H\) is a perfect code of \(G\)? The authors of this paper generalize the result of \textit{J. Zhang} and \textit{S. Zhou} [Eur. J. Comb. 91, Article ID 103228, 10 p. (2021; Zbl 1459.05126)] that if \(G\) is a nilpotent group and \(H\) is a subgroup of \(G\), then \(H\) is a perfect code of \(G\) if and only if the Sylow 2-subgroup of \(H\) is a perfect code of \(G\) for every group with a normal Sylow 2-subgroup.