2021/09/14 by Angelot Behajaina, Roghayeh Maleki, Behajaina, Angelot +3 · 1 citation
Computer Science · Mathematics · #05C25 #05C69 #94B25 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2109.06993
openalex publication_date 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X = (V,E) be a graph. A subset C ⊆ V(X) is a perfect code of X if C is a coclique of X with the property that any vertex in V(X)∖ C is adjacent to exactly one vertex in C. Given a finite group G with identity element e and H≤ G, H is a subgroup perfect code of G if there exists an inverse-closed subset S ⊆ G∖ \e\ such that H is a perfect code of the Cayley graph Cay(G,S) of G with connection set S. In this short note, we give an infinite family of finite groups G admitting a non-normal subgroup perfect code H such that there exists g∈ G with g2∈ H but (gh)2 ≠ e, for all h ∈ H; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. \it arXiv preprint arXiv:2006.05100, 2020].