Permutations as products of k conjugate involutions

Let c(°) denote the number of cycles of a permutation ° of n letters, and let Tr(°)=n-c(°). A t-involution is a product of t disjoint transpositions. Let k>2, n>2t. Theorem. ° is the product of k t-involutions if and only if kt =Tr(°)+2r for some nonegative integer r. (For k=2 one more condition is needed; see [2].) As a corollary, the least power of a class of an involution with at least one fixed point that covers the alternating group is determined.