Diagonal subgroup explained

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

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

This subgroup is isomorphic to

Properties and applications

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

(x _{1,} ... , x _{n)} ⋅ (g, ... , g) = (x _{1} ⋅ g, ... , x _{n} ⋅ g).

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

See also

• Diagonalizable group

References

• .