## Abstract

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 S_n . We prove that if n is large enough, up to isomorphism and duality, the number of string C-groups of rank r for S_n (with

r ≥ (n + 3)/2) is the same as the number of string C-groups of rank r +1 for S_{n+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 S_n 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 S_n is thus reduced to classifying string C-groups of rank r for S_{2r−3} . A consequence of this result is the complete classiﬁcation of all string C-groups of Sn with rank n-κ for κ ∈ {1, . . . , 7}, 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, 135)

This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It is available as sequence number A359367.

r ≥ (n + 3)/2) is the same as the number of string C-groups of rank r +1 for S_{n+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 S_n 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 S_n is thus reduced to classifying string C-groups of rank r for S_{2r−3} . A consequence of this result is the complete classiﬁcation of all string C-groups of Sn with rank n-κ for κ ∈ {1, . . . , 7}, 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, 135)

This sequence of integers is new according to the On-Line Encyclopedia of Integer Sequences. It is available as sequence number A359367.

Original language | English |
---|---|

Number of pages | 49 |

Journal | Advances in Mathematics |

Volume | 453 |

DOIs | |

Publication status | Published - 18 Jul 2024 |

## Keywords

- regular polytope
- string C-group
- symmetric group