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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - F. M. Malyshev
TI  - Realization of even permutations of even degree by products of four involutions without fixed points
JO  - Diskretnaya Matematika
PY  - 2023
SP  - 18
EP  - 33
VL  - 35
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2023_35_2_a1/
LA  - ru
ID  - DM_2023_35_2_a1
ER  - 
%0 Journal Article
%A F. M. Malyshev
%T Realization of even permutations of even degree by products of four involutions without fixed points
%J Diskretnaya Matematika
%D 2023
%P 18-33
%V 35
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2023_35_2_a1/
%G ru
%F DM_2023_35_2_a1
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/

[1] Bourbaki N., Groupes et Algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968, 282 pp. | MR | Zbl

[2] Artin E., Geometricheskaya algebra, Nauka, M., 1969, 288 pp.

[3] Halmos P. R., Kakutani S., “Products of symmetries”, Bull. Amer. Math. Soc., 64 (1958), 77–78 | DOI | MR | Zbl

[4] Radjavi H., “Products of Hermitian matrices and symmetries”, Proc. Amer. Math. Soc., 21 (1969), 369–372 | DOI | MR | Zbl

[5] Sampson A. R., “A note on a new matrix decomposition”, Linear Algebra Appl., 8 (1974), 459–463 | DOI | MR | Zbl

[6] Waterhouse W. C., “Factoring unimodular matrices”, Advanced Problem 5876, Solution, Am. Math. Mon., 81, 1974, 1035 | DOI | MR

[7] Gustafson W. H., Halmos P. R., “Products of involutions”, Linear Algebra Appl., 13:1–2 (1976), 157–162 | DOI | MR | Zbl

[8] Moran G., “Permutations as products of $k$ conjugate involutions”, J. Comb. Theory Ser. A., 19:2 (1975), 240–242 | DOI | MR | Zbl

[9] Borel A., Carter R., Curtis C. W., Iwahori N., Springer T. A., Steinberg R., Seminar on Algebraic Groups and Related Finite Groups, Lect. Notes Math., 131, Springer, 1970, 326 pp. | DOI | MR | Zbl

[10] Petrov N. T., “O dline prostykh grupp”, Dokl. AN SSSR, 208:3 (1973), 537–540 | Zbl

[11] Dénes J., “The representation of a permutation as the product of a minimal number of transpositions, and its connection with the theory of graphs”, Publ. Math. Inst. Hungar. Acad. Sci., 4 (1959), 63–70 | MR

[12] Pikar S., “O bazisakh simmetricheskoi gruppy”, Kiberneticheskii sb. Novaya seriya, v. 1, Mir, M., 1965, 7–34

[13] Kapelmakher V. L., Lisovets V. A., “Posledovatelnoe porozhdenie podstanovok s pomoschyu bazisa transpozitsii”, Kibernetika, 3 (1975), 17–21

[14] Suschanskii V. I., Voskanyan R. A., “O sistemakh porozhdayuschikh simmetricheskikh i znakoperemennykh grupp, sostoyaschikh iz tsiklov odinakovoi dliny”, Voprosy teorii grupp i gomologicheskoi algebry, Yaroslavl, 1985, 43–49

[15] Zubov A. Yu., “O predstavlenii podstanovok v vide proizvedeniya transpozitsii i polnogo tsikla”, Fundamentalnaya i prikladnaya matem., 15:1 (2009), 31–52 | MR

[16] Lugo M., Profiles of large combinatorial structures, PhD Thesis, Univ. Pensylvania, 2010 | MR

[17] Zubov A. Yu., “Krugovye inversii perestanovok i ikh ispolzovanie v zadachakh sortirovki”, Prikladnaya diskret. matem., 31:1 (2016), 13–30

[18] Mikhailov V. G., “Chislo razlozhenii sluchainoi podstanovki v kompozitsiyu dvukh involyutsii s zadannym tsiklom v odnom iz somnozhitelei”, Matem. voprosy kriptografii, 8:1 (2017), 81–94

[19] Bugay L., “Some involutions which generate the finite symmetric group”, Math. Sci. Appl. E-Notes, 8:1 (2020), 25–28 | DOI