Diagonal subgroup

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

[math]\displaystyle{ \{(g, \dots, g) \in G^n : g \in G\}. }[/math]

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 Xⁿ induced by the action of G on X, defined by

[math]\displaystyle{ (x_1, \dots, x_n) \cdot (g, \dots, g) = (x_1 \!\cdot g, \dots, x_n \!\cdot g). }[/math]

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

See also

• Diagonalizable group

References

• Sahai, Vivek; Bist, Vikas (2003), Algebra, Alpha Science Int'l Ltd., p. 56, ISBN 9781842651575, https://books.google.com/books?id=VsoyRX_nHLkC&pg=PA56 .