Derangements in wreath products of permutation groups

2022/07/04 by Arumugam, Vishnuram, Dietrich, Heiko, Glasby, S. P.
#05A05 #20B07 #20B35 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2207.01215

Abstract

Given a finite group G acting on a set X let δ_k(G,X) denote the proportion of elements in G that have exactly k fixed points in X. Let S_n denote the symmetric group acting on [n]=\1,2,…,n\. For A\leS_m and B\leS_n, the permutational wreath product A\wr B has two natural actions and we give formulas for both, δ_k(A\wr B,[m]×[n]) and δ_k(A\wr B,[m]^[n]). We prove that for k=0 the values of these proportions are dense in the intervals [δ₀(B,[n]),1] and [δ₀(A,[m]),1]. Among further result, we provide estimates for δ₀(G,[m]^[n]) for subgroups G≤ S_m\wrS_n containing A_m^[n].

Related

• Products of normal subsets and derangements
• Computing derangement probabilities of the symmetric group acting on\n k-sets
• Derangements in simple and primitive groups
• Derangements in permutation groups with two orbits
• Embedding permutation groups into wreath products in product action