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
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
- 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 proven using the action of the twofold diagonal subgroup.
gollark: Hell is known to be maintained at a temperature of less than something like 460 degrees due to the presence of molten brimstone.
gollark: Despite humans' constant excretion of excess water, holy water levels are actually maintained in the body through the actions of the holicase enzyme.
gollark: I assumed that holy water was some form of metastable state, given that they don't produce it centrally as far as I know.
gollark: Is holiness preserved through evaporation/condensation?
gollark: We're also working on a project to replace iron mines with transubstantiation of wine and iron extraction from hemoglobin.
See also
References
- Sahai, Vivek; Bist, Vikas (2003), Algebra, Alpha Science Int'l Ltd., p. 56, ISBN 9781842651575.
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