On Codes in (Generalized) Symmetric Groups

Abstract

In this article, we show that, for the symmetric group $S_n$, the Young subgroup $S_{\lambda}$, with $l(\lambda)\geq 3$, is not a code with respect to a conjugacy class of $S_n$. This provides a partial answer to \cite[Problem 4.1]{FL26}. We then provide a characterization of conjugacy classes $X_i$, such that subgroups $S_{(n-2,1,1)}$, $S_{(n-3,2,1)}$, and $Y_k$ for $k\leq 3$ are codes for $X_1 \cup X_2$. Furthermore, we describe some codes in the finite Coxeter groups of type $B_n$, $C_n$, $D_n$, and in the generalized symmetric group $C_m \wr S_n$.