Diagonal subgroup

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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\}.} {\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 ) . {\displaystyle (x_{1},\dots ,x_{n})\cdot (g,\dots ,g)=(x_{1}\!\cdot g,\dots ,x_{n}\!\cdot g).} {\displaystyle (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 n. More generally, for an integer k, if G acts kn-transitively on X, G acts k-transitively on X n.
  • Burnside's lemma can be proved using the action of the twofold diagonal subgroup.

See also

References