diagonal subgroup

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

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

This subgroup is isomorphic to {{math|G.}}

Properties and applications

If {{math|G}} acts on a set {{math|X,}} the n-fold diagonal subgroup has a natural action on the Cartesian product {{math|Xⁿ}} induced by the action of {{math|G}} on {{math|X,}} defined by

: $(x_1, \dots, x_n) \cdot (g, \dots, g) = (x_1 \!\cdot g, \dots, x_n \!\cdot g).$

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

See also

Diagonalizable group

References

{{citation|title=Algebra|first1=Vivek|last1=Sahai|first2=Vikas|last2=Bist|publisher=Alpha Science Int'l Ltd.|year=2003|isbn=9781842651575|page=56|url=https://books.google.com/books?id=VsoyRX_nHLkC&pg=PA56}}.

Category:Group theory

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