Normal coverings of finite symmetric and alternating groups

2010/11/19 by Daniela Bubboloni, Bubboloni, Daniela, Cheryl Praeger +1
Mathematics · #20B30 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20B30

paper · pdf · doi:10.48550/arxiv.1011.4368

arxiv created 2010/11/19 · arxiv updated 2010/11/22

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.

Related

• Conjectures on the normal covering number of the finite symmetric and alternating groups
• Normal coverings and pairwise generation of finite alternating and symmetric groups
• Nilpotent covers of symmetric and alternating groups
• Normal covering numbers for S_n and A_n and additive combinatorics
• On the covering number of symmetric groups of even degree