Normal coverings of finite symmetric and alternating groups

TL;DR

This paper studies the minimal number of maximal subgroups needed to cover all elements of symmetric and alternating groups, revealing bounds related to the Euler phi-function and the arithmetical properties of n.

Contribution

It establishes bounds on the minimal covering number for S_n and A_n, linking it to number-theoretic functions and providing exact values in specific cases.

Findings

Number of subgroups between a.phi(n) and bn

Exact covering number for odd n in S_n and even n in A_n

Upper bounds for n divisible by at most two primes

Abstract

In this paper we investigate the minimum number of maximal subgroups H_i for i=1 ...k of the symmetric group S_n (or the alternating group A_n) such that each element in the group S_n (respectively A_n) lies in some conjugate of one of the H_i. We prove that this number lies between a.phi(n) and bn for certain constants a, b, where phi(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+phi(n)/2, and we determine the exact value for S_n when n is odd and for A_n when n is even.