The last patch for classifying shuffle groups

Abstract

Divide a deck of $kn$ cards into $k$ equal piles and place them from left to right. The standard shuffle $\sigma$ is performed by picking up the top cards one by one from left to right and repeating until all cards have been picked up. For every permutation $\tau$ of the $k$ piles, use $\rho_{\tau}$ to denote the induced permutation on the $kn$ cards. The shuffle group $G_{k,kn}$ is generated by $\sigma$ and the $k!$ permutations $\rho_{\tau}$. It was conjectured by Cohen et al in 2005 that the shuffle group $G_{k,kn}$ contains $A_{kn}$ if $k\geq3$, $(k,n)\ne\{4,2^f\}$ for any positive integer $f$ and $n$ is not a power of $k$. Very recently, Xia, Zhang and Zhu reduced the proof of the conjecture to that of the $2$-transitivity of the shuffle group and then proved the conjecture under the condition that $k\ge4$ or $k\nmid n$. In this paper, we proved that the group $G_{3,3n}$ is $2$-transitive for any positive integer $n$ which is a multiple of $3$ but not a power of $3$. This result leads to the complete classification of the shuffle groups $G_{k,kn}$ for all $k\ge2$ and $n\ge1$.