Products of involution classes in infinite symmetric groups

Abstract:

Let $A$ be an infinite set. Denote by ${S_A}$ the group of all permutations of $A$, and let ${R_i}$, denote the class of involutions of $A$ moving $|A|$ elements and fixing $i$ elements $(0 \leqslant i \leqslant |A|)$. The products ${R_i}{R_j}$ were determined in [M1]. In this article we treat the products ${R_{{i_1}}} \cdots {R_{{i_n}}}$ for $n \geqslant 3$. Let INF denote the set of permutations in ${S_A}$ moving infinitely many elements. We show: (1) ${R_{{i_1}}} \cdots {R_{{i_n}}} = {S_A}$ for $n \geqslant 4$. (2)(a) ${R_i}{R_j}{R_k} = \operatorname {INF}$ if $\{ i, j, k\}$ contains two integers of different parity; (b) ${R_i}{R_j}{R_k} = {S_A}$ if $i + j + k > 0$ and all integers in $\{ i, j, k\}$ have the same parity. (3) $R_0^3 = {S_A}\backslash E$, where $\theta \in E$ iff $\theta$ satisfies one of the following three conditions: (i) $\theta$ moves precisely three elements. (ii) $\theta$ moves precisely five elements. (iii) $\theta$ moves precisely seven elements and has order $12$. These results were announced in 1973 in [MO]. (1) and part of (2)(a) were generalized recently by Droste [D1, D2].