Proof of a conjecture of Green and Liebeck on codes in symmetric groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915687273857024 |
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| author | Fang, Teng Li, Jinbao |
| author_facet | Fang, Teng Li, Jinbao |
| contents | Let $A$ and $B$ be subsets of a finite group $G$ and $r$ a positive integer. If for every $g\in G$, there are precisely $r$ pairs $(a,b)\in A\times B$ such that $g=ab$, then $B$ is called a code in $G$ with respect to $A$ and we write $r G=A\boldsymbol{\cdot}B$. 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 $S_n$. Let $n>2k$ and let $j$ be such that $2^j\leqslant k<2^{j+1}$. Suppose that $X$ is a conjugacy class in $S_n$ containing $x$, and $Y_k$ is the subgroup $S_k\times S_{n-k}$ of $S_n$, where the factor $S_k$ permutes the subset $\{1,\ldots,k\}$ and the factor $S_{n-k}$ permutes the subset $\{k+1,\ldots,n\}$. We prove that $r S_n=X\boldsymbol{\cdot}Y_k$ for some positive integer $r$ if and only if the cycle type of $x$ has exactly one cycle of length $2^i$ 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proof of a conjecture of Green and Liebeck on codes in symmetric groups Fang, Teng Li, Jinbao Combinatorics Group Theory Representation Theory Let $A$ and $B$ be subsets of a finite group $G$ and $r$ a positive integer. If for every $g\in G$, there are precisely $r$ pairs $(a,b)\in A\times B$ such that $g=ab$, then $B$ is called a code in $G$ with respect to $A$ and we write $r G=A\boldsymbol{\cdot}B$. 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 $S_n$. Let $n>2k$ and let $j$ be such that $2^j\leqslant k<2^{j+1}$. Suppose that $X$ is a conjugacy class in $S_n$ containing $x$, and $Y_k$ is the subgroup $S_k\times S_{n-k}$ of $S_n$, where the factor $S_k$ permutes the subset $\{1,\ldots,k\}$ and the factor $S_{n-k}$ permutes the subset $\{k+1,\ldots,n\}$. We prove that $r S_n=X\boldsymbol{\cdot}Y_k$ for some positive integer $r$ if and only if the cycle type of $x$ has exactly one cycle of length $2^i$ 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 $G$. |
| title | Proof of a conjecture of Green and Liebeck on codes in symmetric groups |
| topic | Combinatorics Group Theory Representation Theory |
| url | https://arxiv.org/abs/2502.10744 |