Derangements in wreath products of permutation groups

Abstract

AbstractGiven a finite group G acting on a set X let $$\delta _k(G,X)$$ δ k ( G , X ) denote the proportion of elements in G that have exactly k fixed points in X. Let $$\mathcal {S}_n$$ S n denote the symmetric group acting on $$[n]=\{1,2,\dots ,n\}$$ [ n ] = { 1 , 2 , ⋯ , n } . For $$A\leqslant \mathcal {S}_m$$ A ⩽ S m and $$B\leqslant \mathcal {S}_n$$ B ⩽ S n , the permutational wreath product $$A\wr B$$ A ≀ B has two natural actions and we give formulas for both, $$\delta _k(A\wr B,[m]{\times }[n])$$ δ k ( A ≀ B , [ m ] × [ n ] ) and $$\delta _k(A\wr B,[m]^{[n]})$$ δ k ( A ≀ B , [ m ] [ n ] ) . We prove that for $$k=0$$ k = 0 the values of these proportions are dense in the intervals $$[\delta _0(B,[n]),1]$$ [ δ 0 ( B , [ n ] ) , 1 ] and $$[\delta _0(A,[m]),1]$$ [ δ 0 ( A , [ m ] ) , 1 ] . Among further results, we provide estimates for $$\delta _0(G,[m]^{[n]})$$ δ 0 ( G , [ m ] [ n ] ) for subgroups $$G\leqslant \mathcal {S}_m\wr \mathcal {S}_n$$ G ⩽ S m ≀ S n containing $$\mathcal {A}_m^{[n]}$$ A m [ n ] .