Normal covering numbers for S_n and A_n and additive combinatorics

2024/10/09 by Sean Eberhard, Eberhard, Sean, Connor Mellon +1
Computer Science · Engineering · Mathematics · #11P70 #20B30 #20B35 #Advanced Mathematical Theories #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2410.06999

openalex publication_date 2024/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The normal covering number γ(G) of a finite group G is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for γ(S_n) and γ(A_n) depending on the arithmetic structure of n. In particular we determine the limsups over γ(S_n) / n and γ(A_n) / n over the sequences of even and odd integers, as well as the liminf of γ(S_n) / n over even integers. In general we explain how the values of γ(S_n) / n and γ(A_n) / n are related to problems in additive combinatorics. These results answer most of the questions posed by Bubboloni, Praeger, and Spiga as Problem 20.17 of the Kourovka Notebook.

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