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

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

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

${\displaystyle (x_{1},\dots ,x_{n})\cdot (g,\dots ,g)=(x_{1}\!\cdot g,\dots ,x_{n}\!\cdot g).}$

• If G acts n-transitively on X, then the n-fold diagonal subgroup acts transitively on 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.