2023/07/27 by Junyang Zhang, Zhang, Junyang
Computer Science · Engineering · #Algorithms and Data Compression #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2307.15012
openalex publication_date 2023/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Divide a deck of kn cards into k equal piles and place them from left to right. The standard shuffle σ 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 τ of the k piles, use ρτ to denote the induced permutation on the kn cards. The shuffle group Gk,kn is generated by σ and the k! permutations ρτ. It was conjectured by Cohen et al in 2005 that the shuffle group Gk,kn contains Akn if k≥3, (k,n)≠\4,2f\ 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≥4 or k\nmid n. In this paper, we proved that the group G3,3n 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 Gk,kn for all k≥2 and n≥1.