Diagonal subgroup
From HandWiki
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
- [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 X n 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 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
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.
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