Geodesic
The distinguishing number of the direct product and wreath product action.
Journal of Algebraic Combinatorics, Tome 24 (2006) no. 3, pp. 331-345.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $G$ be a group acting faithfully on a set $X$. The distinguishing number of the action of $G$ on $X$, denoted $D _{G}( X)$, is the smallest number of colors such that there exists a coloring of $X$ where no nontrivial group element induces a color-preserving permutation of $X$. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups $G$ and $H$ acting on sets $X$ and $Y$ respectively, we characterize the distinguishing number of the wreath product $G \? _{Y} H$ in terms of the number of distinguishing colorings of $X$ with respect to $G$ and the distinguishing number of the action of $H$ on $Y$. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups $S _{m} \times S _{n}$ on $[ m] \times [ n]$.
Keywords: keywords symmetry group, symmetry breaking, distinguishing number, wreath product, direct product 
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     author = {Chan, Melody},
     title = {The distinguishing number of the direct product and wreath product action.},
     journal = {Journal of Algebraic Combinatorics},
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     number = {3},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2006__24_3_a1/}
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Chan, Melody. The distinguishing number of the direct product and wreath product action.. Journal of Algebraic Combinatorics, Tome 24 (2006) no. 3, pp. 331-345. http://geodesic.mathdoc.fr/item/JAC_2006__24_3_a1/