Diagonal subgroup

In the mathematical discipline of group theory, for a given group $G,$ the diagonal subgroup of the n-fold direct product $G^{&hairsp;&hairsp;n}$ is the subgroup


 * $$\{(g, \dots, g) \in G^n : g \in G\}.$$

This subgroup is isomorphic to $G.$

Properties and applications

 * If $G$ acts on a set $X,$ the n-fold diagonal subgroup has a natural action on the Cartesian product $X^{&thinsp;n}$ induced by the action of $G$ on $X,$ defined by
 * $$(x_1, \dots, x_n) \cdot (g, \dots, g) = (x_1 \!\cdot g, \dots, x_n \!\cdot g).$$


 * If $G$ acts $n$-transitively on $X,$ then the $n$-fold diagonal subgroup acts transitively on $X^{&thinsp;n}.$ More generally, for an integer $k,$ if $G$ acts $kn$-transitively on $X,$ $G$ acts $k$-transitively on $X^{&thinsp;n}.$
 * Burnside's lemma can be proved using the action of the twofold diagonal subgroup.