2022/12/24 by Peter J. Cameron, Cameron, Peter J., Maria Elisa Fernandes +3 · 1 citation
Computer Science · #20B30 #52B11 #Algorithms and Data Compression #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Natural Language Processing Techniques #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2212.12723
openalex publication_date 2022/12/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If G is a transitive group of degree n having a string C-group of rank r≥ (n+3)/2, then G is necessarily the symmetric group Sn. We prove that if n is large enough, up to isomorphism and duality, the number of string C-groups of rank r for Sn (with r≥ (n+3)/2) is the same as the number of string C-groups of rank r+1 for Sn+1. This result and the tools used in its proof, in particular the rank and degree extension, imply that if one knows the string C-groups of rank (n+3)/2 for Sn with n odd, one can construct from them all string C-groups of rank (n+3)/2+k for Sn+k for any positive integer k. The classification of the string C-groups of rank r≥ (n+3)/2 for Sn is thus reduced to classifying string C-groups of rank r for S2r-3. A consequence of this result is the complete classification of all string C-groups of Sn with rank n-κ for κ∈\1,…,6\, when n≥ 2κ+3, which extends previously known results. The number of string C-groups of rank n-κ, with n≥ 2κ+3, of this classification gives the following sequence of integers indexed by κ and starting at κ= 1: (1,1,7,9,35,48) This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It will be available as sequence number A359367.