Commutator subgroup of Sylow 2-subgroups of alternating group and the commutator width in the wreath product

Ruslan V. Skuratovskii

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2020), pp. 3-16.

Résumé

It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i},p_i\in \mathbb{N}$, is equal to $1$. The commutator width of direct limit of wreath product of cyclic groups is found. This paper gives upper bounds of the commutator width $(cw(G))$ [1] of a wreath product of groups. A presentation in the form of wreath recursion [6] of Sylow $2$-subgroups $Sy{l}_2{A}_{2^k}$ of $A_{2^k}$ is introduced. As a corollary, we obtain a short proof of the result that the commutator width is equal to $1$ for Sylow $2$-subgroups of the alternating group $A_{2^k}$, where $k>2$, permutation group $S_{2^k}$ and for Sylow $p$-subgroups $Sy{l}_2{A}_{p^k}$ and $Sy{l}_2{S}_{p^k}$. The commutator width of permutational wreath product $B\wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B\wr C_n$ for an arbitrary group $B$ is found.

Bibliographie

[1] Muranov A., “Finitely generated infinite simple groups of infinite commutator width”, International Journal of Algebra and Computation, 17 (2007), 607–659 | DOI | MR | Zbl

[2] Meldrum J. D. P., Wreath Products of Groups and Semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematic, 1st Edition, 1995, 425 pp. | MR | Zbl

[3] Nikolov Nikolay., “On the commutator width of perfect groups”, Bull. London Math. Soc., 36 (2004), 30–36 | DOI | MR | Zbl

[4] Fite W. B., “On metabelian groups”, Trans. Amer. Math. Soc., 3 (1902), 331–353 | DOI | MR | Zbl

[5] Roger C. Lyndon, Paul E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | Zbl

[6] Nekrashevych V., Self-similar groups, Mathematical Syrveys and Monographs, 117, International University Bremen. American Mathematical Society, 2005 | DOI | MR

[7] Bondarenko I. V., Samoilovych I. O., “On finite generation of self-similar groups of finite type”, Int. J. Algebra Comput., 23:01 (2013), 69–77 | DOI | MR

[8] Grigorchuk R. I., “Solved and unsolved problems around one group”, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects (Bassel, 2005), Progress Math., 248, 117–218 | DOI | MR | Zbl

[9] Lavrenyuk Y., “On the finite state automorphism group of a rooted tree”, Algebra and Discrete Mathematics, 1 (2002), 79–87 | MR | Zbl

[10] Kaloujnine L., “La structure des p-groupes de Sylow des groupes symetriques finis”, Annales Scientifiques de l'Ecole Normale Superieure, 65 (1948), 239–276 | DOI | MR | Zbl

[11] Pawlik B., “The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs”, Algebra and Discrete Mathematics, 21 (2016), 264–281 | MR | Zbl

[12] Skuratovskii R., “Corepresentation of a Sylow p-subgroup of a group $S_n$”, Cybernetics and systems analysis, 2009, no. 1, 27–41 | MR | Zbl

[13] Skuratovskii R., “Generators and relations for Sylow $p$-subgroups of group $S_n$”, Naukovi Visti KPI, 4 (2013), 94–105 (in Ukrainian)

[14] Skuratovskii R. V., Drozd Y. A., “Generators and relations for wreath products of groups”, Ukr. Math. J., 60 (2008), 1168–1171 | DOI | MR

[15] Skuratovskii R. V., “Minimal generating systems and properties of $Syl_2A_{2^k}$ and $Syl_2A_{n}$”, X International Algebraic Conference in Odessa dedicated to the 70th anniversary of Yu. A. Drozd, 2015, 104

[16] Skuratovskii R. V., “Involutive irreducible generating sets and structure of sylow 2-subgroups of alternating groups”, ROMAI J., 13:1 (2017), 117–139 | MR | Zbl

[17] Skuratovskii R. V., Structure and minimal generating sets of Sylow 2-subgroups of alternating groups, arXiv: 1702.05784v2

[18] Skuratovskii R., “Commutators subgroups of sylow subgroups of alternating and symmetric groups their minimal generating sets”, The XII International Algebraic Conference in Ukraine (Vinnytsia, 2019), v. 1, 75 | MR

[19] Dmitruk U., Suschansky V., “Structure of 2-sylow subgroup of alternating group and normalizers of symmetric and alternating group”, UMJ, 1981, 304–312 | MR | Zbl

[20] Guralnick Robert, “Commutators and wreath products”, Contemporary Mathematics, 524, 2010, 97–107 | DOI | MR

[21] Heineken H., “Normal embeddings of p-groups into p-groups”, Proc. Edinburgh Math. Soc., 35 (1992), 309–314 | DOI | MR | Zbl

[22] Ward David, Topics in Finite Groups: Homology Groups, Pi-product Graphs, Wreath Products and Cuspidal Characters, Manchester Institute for Mathematical Sciences School of Mathematics, 2015