Conjugacy classes whose square is an infinite symmetric group

Abstract

Let X ν {X_\nu } be the set of all permutations ξ \xi of an infinite set A A of cardinality ℵ ν {\aleph _\nu } with the property: every permutation of A A is a product of two conjugates of ξ \xi . The set X 0 {X_0} is shown to be the set of permutations ξ \xi satisfying one of the following three conditions: (1) ξ \xi has at least two infinite orbits. (2) ξ \xi has at least one infinite orbit and infinitely many orbits of a fixed finite size n n . (3) ξ \xi has: no infinite orbit; infinitely many finite orbits of size k , l k,l and k + l k + l for some positive integers k , l k,l ; and infinitely many orbits of size > 2 > 2 . It follows that ξ ∈ X 0 \xi \in {X_0} iff some transposition is a product of two conjugates of ξ \xi , and ξ \xi is not a product σ i \sigma i , where σ \sigma has a finite support and i i is an involution. For ν > 0 , ξ ∈ X ν \nu > 0,\;\xi \in {X_\nu } iff ξ \xi moves ℵ ν {\aleph _\nu } elements, and satisfies (1), (2) or ( 3 ′ ) (3’) , where ( 3 ′ ) (3’) is obtained from (3) by omitting the requirement that ξ \xi has infinitely many orbits of size > 2 > 2 . It follows that for ν > 0 , ξ ∈ X ν \nu > 0,\;\xi \in {X_\nu }\; iff ξ \xi moves ℵ ν {\aleph _\nu } elements and some transposition is the product of two conjugates of ξ \xi . The covering number of a subset X X of a group G G is the smallest power of X X (if any) that equals G G [AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.