2025/02/15 by Teng Fang, Fang, Teng, Li, Jinbao · 1 citation
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #graph theory and CDMA systems #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.2502.10744
Let A and B be subsets of a finite group G and r a positive integer. If for every g∈ G, there are precisely r pairs (a,b)∈ A× B such that g=ab, then B is called a code in G with respect to A and we write r G=A\boldsymbol⋅B. If in addition B is a subgroup of G, then we say that B is a subgroup code in G. In this paper we resolve a conjecture by Green and Liebeck \cite[Conjecture 2.3]Green20 on certain subgroup codes in the symmetric group Sn. Let n>2k and let j be such that 2j\leqslant k<2j+1. Suppose that X is a conjugacy class in Sn containing x, and Yk is the subgroup Sk× Sn-k of Sn, where the factor Sk permutes the subset \1,…,k\ and the factor Sn-k permutes the subset \k+1,…,n\. We prove that r Sn=X\boldsymbol⋅Yk for some positive integer r if and only if the cycle type of x has exactly one cycle of length 2i for 0\leqslant i\leqslant j and all other cycles have length at least k+1. We also propose several problems concerning the existence of certain subgroup codes in a finite group G with respect to a conjugation-closed subset in G.