Geodesic
Realization of even permutations of even degree by products of four involutions without fixed points
Diskretnaya Matematika, Tome 35 (2023) no. 2, pp. 18-33

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We consider representations of an arbitrary permutation $\pi$ of degree $2n$, $n\geqslant3$, by products of the so-called $(2^n)$-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four $(2^n)$-permutations. Products of three $(2^n)$-permutations cannot represent all even permutations. Any odd permutation is realized (for odd $n$) by a product of five $(2^n)$-permutations.
Keywords: alternating group, permutation, involution, generator, cyclic structure, length of an element of a group. 
@article{DM_2023_35_2_a1,
     author = {F. M. Malyshev},
     title = {Realization of even permutations of even degree by products of four involutions without fixed points},
     journal = {Diskretnaya Matematika},
     pages = {18--33},
     publisher = {mathdoc},
     volume = {35},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2023_35_2_a1/}
}
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F. M. Malyshev. Realization of even permutations of even degree by products of four involutions without fixed points. Diskretnaya Matematika, Tome 35 (2023) no. 2, pp. 18-33. http://geodesic.mathdoc.fr/item/DM_2023_35_2_a1/