Geodesic
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, Tome 21 (2016) no. 2, pp. 264-281.

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

Base (minimal generating set) of the Sylow 2-subgroup of $S_{2^n}$ is called diagonal if every element of this set acts non-trivially only on one coordinate, and different elements act on different coordinates. The Sylow 2-subgroup $P_n(2)$ of $S_{2^n}$ acts by conjugation on the set of all bases. In presented paper the stabilizer of the set of all diagonal bases in $S_n(2)$ is characterized and the orbits of the action are determined. It is shown that every orbit contains exactly $2^{n-1}$ diagonal bases and $2^{2^n-2n}$ bases at all. Recursive construction of Cayley graphs of $P_n(2)$ on diagonal bases ($n\geq2$) is proposed.
Keywords: Sylow $p$-subgroup, group base, wreath product of groups, Cayley graphs. 
@article{ADM_2016_21_2_a7,
     author = {Bart{\l}omiej Pawlik},
     title = {The action of {Sylow} 2-subgroups of~symmetric groups on the set of bases and the problem of~isomorphism of their {Cayley} graphs},
     journal = {Algebra and discrete mathematics},
     pages = {264--281},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a7/}
}
TY  - JOUR
AU  - Bartłomiej Pawlik
TI  - The action of Sylow 2-subgroups of~symmetric groups on the set of bases and the problem of~isomorphism of their Cayley graphs
JO  - Algebra and discrete mathematics
PY  - 2016
SP  - 264
EP  - 281
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a7/
LA  - en
ID  - ADM_2016_21_2_a7
ER  - 
%0 Journal Article
%A Bartłomiej Pawlik
%T The action of Sylow 2-subgroups of~symmetric groups on the set of bases and the problem of~isomorphism of their Cayley graphs
%J Algebra and discrete mathematics
%D 2016
%P 264-281
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a7/
%G en
%F ADM_2016_21_2_a7
Bartłomiej Pawlik. 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, Tome 21 (2016) no. 2, pp. 264-281. http://geodesic.mathdoc.fr/item/ADM_2016_21_2_a7/

[1] A. Bier, V. Sushchansky, “Kaluzhnin’s representations of Sylow $p$-subgroups of automorphism groups of $p$-adic rooted trees”, Algebra Discrete Math., 19:1 (2015), 19–38 | MR | Zbl

[2] D. Gorenstein, Finite Groups, Harper's series in modern mathematics, Harper Row, New York, 1968 | MR | Zbl

[3] R. I. Grigorchuk, V. V. Nekrashevych, V. I. Sushchanskii, “Automata, Dynamical Systems, and Groups”, Proc. Steklov Inst. Math., 231 (2000), 134–214 | MR | Zbl

[4] L. Kaluzhnin, “La structure des $p$-groupes de Sylow des groupes symetriques finis”, Ann. Sci. l'Ecole Norm. Sup., 65 (1948), 239–272

[5] B. Pawlik, “Involutive bases of Sylow 2-subgroups of symmetric and alternating groups”, Zesz. Nauk. Pol. Sl. Mat. Stos., 5 (2015), 35–42

[6] V. Sushchansky, A. Słupik, “Minimal generating sets and Cayley graphs of Sylow $p$-subgroups of finite symmetric groups”, Algebra Discrete Math., 2009, no. 4, 167–184 | MR | Zbl