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

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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/}
}
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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/