Realization of permutations of even degree by products of three fixed-point-free involutions

We consider representations of permutations of even degree by products of three pairwise-cycle permutations, all of whose cycles have length 2. Two such factors are not enough for cardinality reasons. The first approaches to solving this problem were undertaken by the israeli mathematician G. Moran in 1975. In 1978, the american algebraist J.L. Brenner gave examples of permutations that cannot be represented as a product of three pairwise-cycle permutations, and proved that any even permutation is obtained as a product of four such permutations. Cardinality considerations suggested that sets of exceptional permutations that are not representable by three factors should be quite observable.
In 1988, G. Moran confirmed this, gave an exhaustive list of exceptional permutations, but acting on infinite sets, with a promise to solve this problem for permutations of finite degree, which has not yet happened. In 2010, the Russian cryptographer M. E. Tuzhilin gave computer-generated lists of exceptional permutations of degree not exceeding 10.
The talk will present a constructive proof of obtaining almost all permutations of the same parity with their degree by products of three pairwise-cycle permutations. The lists of permutations that cannot be represented in this way are explicitly indicated. The problem under consideration on finite sets turned out to be an order of magnitude more complicated than its analogue on an infinite set.