Diagonal subgroup

In the mathematical discipline of group theory, for a given group Template:Math the diagonal subgroup of the n-fold direct product Template:Math is the subgroup

<math>\{(g, \dots, g) \in G^n : g \in G\}.</math>

This subgroup is isomorphic to Template:Math

Properties and applications

• If Template:Math acts on a set Template:Math the n-fold diagonal subgroup has a natural action on the Cartesian product Template:Math induced by the action of Template:Math on Template:Math defined by

<math>(x_1, \dots, x_n) \cdot (g, \dots, g) = (x_1 \!\cdot g, \dots, x_n \!\cdot g).</math>

• If Template:Math acts Template:Math-transitively on Template:Math then the Template:Math-fold diagonal subgroup acts transitively on Template:Math More generally, for an integer Template:Math if Template:Math acts Template:Math-transitively on Template:Math Template:Math acts Template:Math-transitively on Template:Math
• Burnside's lemma can be proved using the action of the twofold diagonal subgroup.

See also

• Diagonalizable group

References

• Template:Citation.

Template:Group-theory-stub