Conjugacy Classes Whose Square is an Infinite Symmetric Group

Abstract

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