Products of involution classes in infinite symmetric groups

Let A be an infinite set. Denote by Sa the group of all permutations of A, and let R, denote the class of involutions of A moving A elements and fixing i elements (Formula presented) The products R(Rj were determined in [Ml]. In this article we treat the products (Formula presented) Let INF denote the set of permutations in Sa moving infinitely many elements. We show: (Formula presented) contains two integers of different parity; (Formula presented) and all integers in (Formula presented) have the same parity.(Formula presented) satisfies one of the following three conditions: 9 moves precisely three elements. 9 moves precisely five elements. 9 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 [Dl, D2].