Diagonal subgroup

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, \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}, \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

Diagonalizable group

References

Sahai, Vivek; Bist, Vikas (2003), Algebra, Alpha Science Int'l Ltd., p. 56, ISBN 9781842651575.

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