Diagonal subgroup
In the mathematical discipline of group theory, for a given group G, the diagonal subgroup of the n-fold direct product G n is the subgroup { ( g , … , g ) ∈ G n : g ∈ G } . {\displaystyle \{(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 n induced by the action of G on X, defined by ( x 1 , … , x n ) ⋅ ( g , … , g ) = ( x 1 ⋅ g , … , x n ⋅ g ) .